There are two responsa in question. The first, in Volume X, Part 25, Chapter 26, Section 6, is the end of a responsum dealing with transplants, particularly heart transplants. In this section, Rabbi Waldenberg sets out to deal with "other significant/organic alterations of the body, such as a person who changes from male to female, or vice versa." He mentions that such surgery is done in special cases, adding the comment, "(rare, of course)".
Rabbi Waldenberg brings sources from centuries ago of women who changed into men. Exactly what the phenomenon in question was isn't clear from what he writes, but it is clear that to the naked eye, a person or persons who were female by appearance became male by appearance.
Discussing whether such a person, if married, would require a writ of divorce from his/her husband, one of the sources cited by Rabbi Waldenberg writes that he/she does not, "because this woman has many signs of being a man which are apparent to the visual sense, she does not require a writ of divorce, because she is truly a man." The source continues to say that in the morning blessings, where a man says, "Blessed are You, Lord our God, King of the Universe, Who has not made me a woman," such a man should end the blessing instead, "Who has changed me into a man" (Rabbi Waldenberg states elsewhere that a male who becomes female would say, "Who has changed me according to His will," rather than the standard, "Who has made me according to His will," which is said by women).
Rabbi Waldenberg discusses the strange case mentioned in the Bible, of Elijah the prophet ascending to heaven. According to Jewish tradition, Elijah was transformed into an angel. During the middle ages, the question was asked whether Elijah's wife would have been permitted to remarry, since her husband had neither died nor divorced her. The conclusion was that since a woman is barred from remarrying so long as she is a man's wife, and in this case she was no longer a man's wife, but an angel's wife (a status which doesn't exist in Jewish law), her marriage would be automatically nullified. Rabbi Waldenberg states that the same would be the case if a man becomes a woman, as the wife of a woman is not a recognized status in Jewish law.
Important elements of this responsum are that a change of sex which results in the individual appearing mostly of a new gender actually changes that individual's gender in the eyes of Jewish law. And that this is obvious enough that it can end a marriage without either death or divorce, which is an extreme position in Jewish law. Anyone of lesser stature than Rabbi Waldenberg would be unlikely to get away with such an opinion.
The second responsum, which is dated 1971 (I don't have the date for the first one, but the second one refers to it), is in Volume XI, Part 78. It begins with a letter sent to Rabbi Waldenberg by a doctor who read the first responsum and wanted to ask about a case which occurred in the hospital where he worked. The case seems to have been one of Androgen Insensitivity Syndrome. A child was born mostly female in appearance. Labia and what seemed to a clitoris were present. But there was a single testicle in one of the labia. When tested, the baby was found to be XY. Genetically male. A invasive examination was performed, and there were no male organs inside the baby's body.
The doctor's two questions were whether it was permissible operate on the child in order to make it functionally female when it was genetically male, and whether it was permissible, in light of the prohibition of castration, to remove the single testicle.
Rabbi Waldenberg begins by noting that the fact that there were no male organs inside the infant's body is irrelevant. "The external sexual organs of the newborn in question, as you have described, appear as those of a female, and it has no external indications of male organs. Only the special examinations [gene testing] showed that male cells were present. And therefore, in my opinion, even if we were to leave it as it is, it would have the status of a female, "since the external organs which can be seen by the naked eye are the determinant in Jewish law."
There is more to the responsum, including a discussion of whether the prohibition of castration applies when the testicles are not functional (there are views in both directions), and I should emphasize that the significance of these responsa applies only to post-op transexuals. There is no implication that actu- ally undergoing hormone therapy and surgery are permissible.
It would be interesting to compare this to Rabbi Waldenberg's responsa regarding abortion. Jewish law is not pro-choice, as that is generally understood; nor is it pro-life, as that is generally understood. Rather, it is considered to be highly undesireable, and possibly close to bloodshed. Nevertheless, the mother's life always takes precedence, and when there is risk to her, rabbis will almost always permit abortion.
Rabbi Waldenberg recognized psychological factors as real ones. He recognized psychological trauma as a danger to a woman which could be grounds for permitting abortion even when no physical danger existed. In light of this, and in light of the terribly high rate of clinical depression and/or suicide among transexuals, a case might be made for permitting hormone therapy and surgery in certain cases. This is conjecture, however, and I don't believe it has been addressed in those terms and in context of Rabbi Waldenberg's determination that surgery actually results in a gender change.