So, given that this is the Judaism DIR, and I would not wish to offer undue objection to my Orthodox colleagues here, I will not presume to offer what I believe would be the Orthodox spectrum of interpretations, given that I am not Orthodox, and do not agree with many of the readings of verses such as these in Orthodoxy today, and so would inevitably bias the presumed interpretation of a specific text; so I will leave that part of the answer to one of the very competent Orthodox members here to do.
As for non-Orthodox Judaism, these are regarded as extremely problematic verses. In the Conservative movement, those who would seek to effectively nullify the effect in Jewish Law of these verses have only been incompletely able to do so. Various interpretations have been made to suggest that the verses only actually are concerned with male-on-male anal sex, and all other forms of same sex relations are permissible. But many feel that these solutions are still deeply insufficient, as they leave gay sexuality with a limitation far graver than any placed on heterosexual relations in the halakhah (Jewish Law), and this still carries a stigma. These insufficiencies mostly result from the fact that the halakhah makes it extremely, extremely difficult to legislate an effective reversal of a negative commandment in the Torah. The one true means to do so has not yet been employed, because of its radical nature, although a small group of rabbis are preparing to do so within the next year or two. This would involve a legislated injunction, known as a takkanah l'akor davar min hatorah (a "legislated injunction to uproot something from the Torah"), though nothing is actually removed from the Torah, of course. What would happen is a legislation by a rabbincal court would dictate that the verses in question are not to be acted upon, and may not be used as prooftexts to support any matter of halakhah, for a duration of 1000 years or until the messiah comes (whichever is first). The stated reason would be that, since it is clear that we cannot find a way to interpret these verses that does not result in a tenth of the Jewish people being permanently in violation of the law for no reason other than the way they were created, it is clear that whatever the proper interpretation of the verses are, we do not know them, and cannot seem to figure out what they are. Since we have not been able to do so for 2000 years or more, this means that we must surely wait for the messiah to come and reveal the true meaning of the verses, since surely God would never command us to oppress our own, based solely on how He created them.
The Reform movement, who feel that they are not strictly bound by halakhah, has come up with various other ways of dealing with these verses, ranging from the singularly uncreative and entirely untraditional method of simply ignoring them without any supporting methodology, to dubious and fairly unconvincing historical guessing games, to various textual interpretations that, while not supported enough by traditional methods to be halakhically acceptable, are still ingenious and promising.
For example, some have argued that the phrase in 18:22 v'et zachar lo tishkav mishkevei ishah ("You must not lie carnally with a male in the way of lying carnally with a woman"), which is phrased in a unique and peculiar structure (the particular wording is used only there, and in 20:13, and nowhere else in the Tanakh), merely indicates that a man ought not to have sexual relations with another man in the same way that he would with a woman. A gay man will never approach sex with a woman with the same naturalness and ease that he would with a man, and vice-versa: thus there is no transgression. It is only a straight man who has sex with another man for reasons other than he would have sex with a woman who has transgressed. So, for example, prison rape would be prohibited, because it is not the act of a gay man who has sex with another gay man for love and mutual pleasure, but the act of a straight man who asserts his dominance over another man by brutalizing him with a violent sex act. It is my belief that, while inspired, this reading cannot be supported by the text using the classical methods of halakhic reading. At least not yet.
Another ingenious example of interpreting these verses is the idea that, because both of these commandments are specifically bracketed by verses prohibiting idolatry and foreign worship, after the fashion of the Egyptians and Canaanites, that what is meant is not any homosexual act, but specifically homosexual acts done in the context of idolatrous worship. That it is only if one were having gay sex as a form of worship of, say, Amen-Ra, or Baal, that it would be forbidden. This still, IMO, is not strongly enough supported for a halakhic reinterpretation, but it is close. If more research could be done, perhaps some classical scholars' opinions found who read the text similarly, maybe this could solve the problem for halakhically observant Jews. But none has, yet. So far, readings like these, despite their striking originality and cleverness, are only useful to Reform and Reconstructionist and Renewal Jews, who do not see themselves as bound by the rules of halakhah.
To my mind, it is deeply important that Judaism not stigmatize and condemn a tenth of its own people for what they cannot control; and it is also important that the issue be dealt with in a way that respects the tradition even while instituting changes. Which, in this case, is difficult. I think that these two verses are the most challenging in the entire Torah, and while I have no proof of it, and certainly no halakhic way to say so or support it, I am sure that these two verses are somehow errors, mistakes the prophets made in understanding what God was trying to convey. I refuse to believe that the God I believe in would create people to be gay and then condemn them for it. The God I believe in is merciful, and forgiving, and just, and compassionate, and does not create people merely to torment them.
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