Very early in your lifetime, you had not yet constructed your immune system. If the embryo that hatches from the zona pellucida happens to be too close to another embryo hatching from a separate zona pellucida, the two could merge and the blastocystic cavity would be built from stem cells having two different nuclear DNA signatures. Then, this mix of stem cells would give rise to the embryoid body and the umbilical vesicle and the immune system molecular markers would be extended to recognize two specific genome origins, not just one.
Jane's body would not reject her own tissues or her twin's tissues, and the embryonic individual (Jane's child) actually fools the immune system of the mother in order to avoid tissue rejection (this is accomplished in three ways that I won't go into unless you're really interested in being bored).
The important point to note is that Jane's twin was not an identical twin (as in having her origin from the same zygote that Jane derievd from). The twin started her lifetime as a separate 'conceptus' with a separate outer coat that protected her on her journey to the uterus. It was when the two embryos hatched from their 'zona pellucidas' that the merging happened, but the merger happened before either embryo had developed an immune system.
It is also possible that Jane's twin was conceived a day or so later than Jane so that when the twin hatched from her zona pellucida, Jane's more robust outer membrane absorbed the just hatched twin and incorporated the twin's genome into the construction of immune system and organs while cancelling the twin's stem cells tasked to build the twin's placental organ.
Jane's trophoblast stem cells (the stem cells tasked with building the placenta and sending chemical signals to bring about implantation in the mother's uterine lining) may have dissolved the primitive trophoblasts of the twin (the outer cells encapsulating the blastocystic cavity where the amniotic sac and embryoid body will grow), then incorporated into Jnae's balstocystic cavity the blastocyctic stem cells of the twin. That is in fact what I would suspect happens in 'traditional' chimeric development. ... A researcher in Chicago (professor Norbert Gleicher from Yale) actually purposely injected male embryo blastocystic stem cells into the blastocystic stem cells of a female embryo, to watch how the process of merging may occur. The chimera embryo was euthanized within a matter of days, to dissect the stem cell mass and see which portions carried the Y chromosome 'introduced' into the female embryo.