Below you will find two sections. The first consists of several comments sent in by readers, which I found helpful. Following that, you will find my proposed resolution of this question.
Selected Comments and ideas that accrued in this discussion, for all of which I thank you:
I Why does achdus have to mean social mixing and true respect? When the old yerushalmi yid trying to buy a dirah for his youngest daughter knocks on my door and I listen to his problems and give him a few dollars, I feel some degree of achdus with him even though we have little shared interests and diametrically opposed ideas. When I see an Modern Orthodox couple, stuck with a child in Mount Sinai hospital for shabbos, enjoying the kugels and cakes provided by Satmar bikkur cholim I have a sense of seeing true achdus.
Achdus, as cliched as it sounds, means treating every Jew as I would my brother. I can (and do) disagree with almost everything my brother says but I would do anything for him and vice versa.
II One of the most commented-upon cases of achdus was the chever atzabim in the time of Achav - which, applying your hypothesis, was made possible by the avodah zarah attitude of tolerance. That tolerance even embraced Ovadiah hiding the true nevi'im. Imagine the converse: how much tolerance a chareidi society would show for someone protecting nevi'ei haba'al.
I once heard in the name of the Chasam Sofer that the gematria of Verav Shalom is the same as Machlokes. It actually works out precisely.
Sholom Rav le'Ohavei Torasecha, indeed.
III I would suggest that in light of Jewish history of the last several millenia perhaps we could define it as uniformity of purpose. Thus Satmar and Zionist messianists do display achdus as their common goal is the establishment of a Torah based religious community despite differing definitions and visions of that community.
IV I have been curious for a long time as to how is it that we say that it was the lack of achdus that caused the 2nd churban (well sinas chinam) and the big 3 were the cause of the 1st churban. After all, there seemed to be no shortage of the big 3 during the bayis sheni as evidenced by the existence of the Baryonim, Misyavnim et al and the first bayis we had 2 separate kingdoms which regularly fought one another in addition to their proclivity toward murder, theft and idolatry.
V Shared destiny, shared basic philosophy (rachmanim, bayshanim gomlei chasadim) despite wide divergence, mystical one-ness (areivus), love at arm's length with occasional aliyah laregel convocation, spiritual connection as evidenced by a guaranteed share in olam haba, shared methodology in seeking answers.
This is how I understand it.
The great sin of the Bayis Sheini was Sinas Chinam. Chinam Davka! Of course there are justifications for hating a fellow Jew. But you have to have clear and valid reasons for hating him. If your reasons are not legally defensible, you will transgress the Mitzva De'Oraysa of Ve'ahavta, the Lahv of Lo sisna es achicha bilvavecha (as is clear in Pesachim 113b, the issur applies even where you tell him you hate him, lahv davka bilvavecha), and the din of being Dan es kol adam le'kaf zechus. What, then, is a valid reason to hate a fellow Jew?
Chazal talk of the word "sonei" in the Torah, in the parshios of "ha'achas senu'ah" and "chamor sona'acha." In both cases, Chazal (Pesachim 113b, Rav Nachman bar Yitzchak,) tell us that it refers to a Jew who is an avaryan, whom you have good reason to hate. In fact, then it would be a mitzva to hate him (Shulchan Aruch CM 34, Rambam 13 Rotzei'ach 14 and 5 Avodah Zara 4-- "lo soveh lo" by a Meisis, which the Rambam learns to mean "don't be his Oheiv.") But, and this is important, remember the magnificent Tosfos in Psachim 113b. Tosfos asks, if Sona'acha refers to a person whom the Torah requires that you hate, why does the Gemara in Bava Metzia say "Oheiv lifrok ve'sonei lit'on, mitzva be'sonei lit'on, ke'dei lakuf es yitzro:" but if you are supposed to hate him, what does lakuf es yitzro mean? Tosfos answers that we are talking about a three-step process here. 1: you hate him, and that's fine, but he sees you hate him, and "kamayim hapanim lapanim," and so 2. he hates you too, and, reacting to his hatred of you, 3. you hate him even more than the "shiur" of hatred that is legitimate. This, Tosfos says, is the work of the Yetzer Hara, and it is a mitzva to help the bad guy just to break your excessive hatred. So even when there is a mitzva to hate him, there is a shiur. The Rambam mentioned above, in 13 Rotzei'ach, also holds like Tosfos, since he writes the din of Lakuf es Yitzro in immediate contiguity to the mitzva of hating an avaryan that refuses to do teshuva. Or, as Artscroll loves to say, he juxtaposes them.
But the great and inadequately examined question is: What is the shiur? I don't recall seeing this discussed in the Steipler's Shiurin shel Torah. Are there different shiurim in Sin'ah? Are there different kinds of sin'ah? I would suggest that, as said in a comment, that the shiur is the amount you would hate your brother if he was an avaryan. I'm not a therapist, so I can't quantify what that means; but it is something that bears consideration. If someone's brother was a criminal, would he not love him? If someone's brother was a traitor, or had done to'eivos, disgusting things, I think the normal reaction would be to try to help him change. You wouldn't want to spend time with him socially, but you would try to help him, or at least wish he could be helped to redeem himself. This is pshat in the passuk that says "lo sisna es achicha bilvavecha." The Torah is teaching that the reason you should eschew hatred is because he is your brother; even when there is a din of sin'ah, you have to realize that you are hating your brother, and it should pain you to have come to such a state. (From Rav Ephraim Greenblatt; Rivevos Ephraim OC 2:198.)
So; the sin of sin'as chinam is a predisposition to indifference or dislike of a fellow Jew. If you have a valid justification for hatred, it is not a sin; by the Bayis Rishon, the division of Klal Yisrael was, I imagine, justified by some halachic logic, though evidently it turned out to be inadequate. According to their analyses, they were fulfilling the dictum of Chazal in Avos DeRav Nassan, end of Perek 16. The fact that what they did seems, to us, indefensible, may be because the winners wrote the history books; that is, the Aseres Hashvatim may have had a mesora in Torah and Halacha that died with them, and which has no vestige in our halacha. (Just for fun, if you wouold gather the oddest shittos of Da'as Yachid in the Shas that we have, both in dinim De'oraysa and dinim De'rabanan, from Rebbi Eliezer to Rav Preida and Rav Chidka, and imagine a Jewish world in which they were accepted as normative halacha, it wouldn't look at all like Judaism as we know it.)
If the other person has a legitimate halachic basis for his crazy shittah, then you have no defensible reason to hate him. According to his mesorah, according to his sincere interpretation of the Torah, his shittah has validity. Is eilu ve'eilu elastic enough to cover Mossad Harav and Satmer? Yes, it is, if only in the sense that both seek the truth through a sincere investigation of our Torah, based on the 13 Ikrim and the 13 Middos She'hatorah Nidreshes Bahem. And that is enough, as indicated in the Rambam in Cheilek which I bring toward the end of this post. So you can't hate him-- but you certainly can, and should, fight him, as Beis Shamai fought Beis Hillel, as Shaul fought David, as the enemies of David fought him, as Yeravam and Rechavam fought, and the fight might be violent. Machlokes is characteristic of the Human condition, and is an inevitable and perhaps even necessary part of being a partisan of your derech of avodas Hashem. But while you fight, you have to remember that the other person is entitled to his opinion. You fight, you try to win, you can despise the shittah he espouses. But you have to remember that he is entitled to that shittah. David Hamelech almost never spoke of Shaul with anything but the greatest respect-- except one time, (Shmuel II:22- Mikaf kol oyvav umi'kaf Shaul) and Hashem rebuked him for that one slip (Shigayon leDavid ahl divrei Kush ben Yemini, and see Moed Kattan 16b). And, you will enjoy the Abarbanel's remarkable interpretation of this passuk, that accents the requirement that a Jew respect and love a fellow Jew, even his mortal enemy.
As one commenter tells us, the Chasam Sofer pointed out that the gematriya of "verav shalom" is the same as that of "machlokes."
The Mitzva of Achdus is an outgrowth of the concept of Areivus. Areivus stems from the words "Kol Yisrael Areivim Zeh Bazeh," although it's usually quoted zeh lazeh. It doesn't matter; the idea is correct no matter whether you say bazeh or lazeh. The word Areivim means commingled, and is similar to the word Areiv as in Arvus, the co-signer of a loan who accepts personal responsibility for the borrower's liability.
The Yerushalmi says that if a fellow Jew has not fulfilled his mitzvah, you can be motzi him with a bracha, because so long as he hasn't done the mitzva, you haven't fulfilled the mitzva completely either, and YOU are a bar chiyuva. Areivus is not only a din in mitzvos. It is a description of our essential reality; Klal Yisrael is intimately intraconnected, and the chesronos of one are the chesronos of all, just as the zechusim of each of us are shared. This is why it is better to daven on Yamim Nora'im with a minyan, because as a group, we share the group's special kesher with Hashem. Matan Torah required that absolute areivus-- Vayichan Yisrael. When you daven with a minyan, your tefilla is heard far above where it would be heard if you davened yourself.
We share our fate. The actions of each effects the other. There is no point in isolating another person or group, because their behavior reflects upon us no matter how we would like to ignore it. Hashem looks upon us as one nation, and our antipathy towards those we dislike does not mitigate or ameliorate that shared judgment.
Our actions stem from a shared history and, hard as it may be to believe, some degree of shared philosophy.
We are all parts of the same body, and you can't despise a part of your own body. If it threatens your life, or it causes terrible pain, you might have to remove it. But amputation is a painful and tragic last resort.
The Gemara in Chagiga 26a says that although during most of the year restrictions on contact with ignorant people were in place, most of these restrictions were lifted on the Shalosh Regalim. The Gemara darshens a passuk of Kol Yisrael chaverim-- on Yomtov, we must act as if all Jews are Chaverim, i.e., respected and scholarly members of our community. As soon as Yomtov was over, though, the real Chaverim were tovel to remove any tumah they may have gotten from the unlearned. And beyond Amei Ha'aretz-- during the year, there were numerous social restrictions that separated various groups of Jews: Bigdei ochlei chulin al taharas truma are medras le'ochlei chulin al taharas hakodesh. You'd want to stay out of arm's length! But for Yomtov, at least, as hard as it must have been for the Prushim, any concern of this tumah was to be disregarded. The vitally important awareness of interconnectedness mandates a kula in Tahara.
As I mentioned above, one of the great questions is what the rishonim mean by saying that despite the mitzva to hate an avaryan, there is still a din of kedei lakuf es yitzro. There is one more question that has elicited a great deal of discussion, and that is the following:
I mentioned the Rambam in Rotzei'ach that says that the mitzva to hate an avaryan applies to one who has been seen to do an aveira be'meizid. However, in his Pirush Hamishnayos in Chelek, Sanhedrin perek 10, in his discussion of the last of the 13 Ikrim, he says the following:
"When a person believes all of these Foundations... he enters into Klal Yisrael and it is a mitzva to love him and to have mercy on him...even if he has done any number of sins because he was driven by his Yetzer Hara and his lust and his vulgar desires; he will be punished...but he has a share in Olam Haba...but if a person has "niskalkeil" one of these Foundations, ...he is separated from the Klal...and is a Min...and it is a mitzva to hate him...and upon him says the passuk (Tehillim 139) Halo mi'san'echa Hashem esna."
As Rabbi Shragie Neuberger once said, once he found that the Rambam said anything about a given subject, he was confident that he could say a lomdishe drasha on that topic. The Rambam never said anything without immediately creating an entire spectrum of bewilderments. This Rambam is a good example.
1. He contradicts his halacha in Rotzei'ach regarding hating an avaryan.
2. He implies that "yeish lo chelek le'olam haba" and the mitzva of "ve'ahavta" are a tautology, that they are intimately related, that either both applicable or neither is. Why would the underlying logic of one be the same as for the other?
3. What does he mean by Niskalkel? An innocent error? Of course, nebach an apikores is also an apikores. But here, he implies that you can't even say nebach.
1. For any Lubavitchers that are reading this: The Tanya (32) says that even when one must hate an oveir aveiros, that only means that you hate the bad in him, but you must also love the good in him, while hatred for Kofrim is absolute-- "tachlis sin'a si'nei'sim." There are those that use this to distinguish between the Rambam in the Yad and in the Pirush Hamishnayos, but to me, it's just wishful thinking.
2. The Yerei'im in 39 says that the sin'a to an oveir aveiros who doesn't respond to tochacha is not "sin'as haleiv, ella sin'a hanir'is le'einayim," and he shtells tzu the Gemara in Pesachim. Also used to be meyasheiv the Rambam, also bubbeh maises.
3. For a thorough discussion of to whom the mitzva of leha'chayoso applies, and the connection with the word "achicha," and whether there is a difference between doing an aveira once of repeatedly and le'tei'avon, and whether there is an in between madriega of "not hating and no chiyuv to support but allowed to support," which would be like "lo moridin aval im ratza, ma'aleh," see the Shach and the Taz in YD beginning of 251.
To be continued.