Linear-Feedback Shift Register (aka LFSR)
Numbers 3 and 7 may seem a little strange in this context, but, let's not declare an idea impossible before thinking about it for five minutes*. There actually is a simple way to arrange triplets of items into a seven-step cycle:
That's a queue of three bits, supporting a "shift" operation in which the rightmost bit is discarded and a new bit is pushed in from the left. The new bit depends on two specific bits of the previous state: If they were different the new value will be 1, and vice versa. This produces a sequence of length 7 before the triplets start to repeat (unless you start from "000", which cycles immediately). The complete sequence is left as exercise to the reader (hope the reader will not find that I messed it up).
The "bits" could be chromozomes, each of them being either X or Y (just different symbols for 0 and 1). Or maybe DNA strands with a polarity, direction? The LFSR thing, then, is their way to scramble genetic information in order to create some novelty in the offspring. Terestrial organisms do have such mechanism, which ensures that chromozomes are not just cloned from parents to children, albeit in a new combination; Instead, parts of a chromozome are mixed together, and with help of some smart evolutional engineering they remain viable. In case of phemogians, the "rightmost two" chromozomes are blended to form a new one, whose polarity/XY-ness is determined by the combination (I guess the three chromozomes in a triploid cell do not have an equal status, so it can be defined which ones are the rightmost). This "meiosis" needs to happen three times in order to build three new chromozomes, with some handwavy molecular compatibility involving the "leftmost chromozome" to enforce the LFSR rule.
It also probably means that the child sex is fully determined by the combination of parents; So, a Sil, a Trem, and a Kiim can only spawn Jeks.
Some remaining questions: Why would evolution bother with this? Seems tricky, but anyway, evolution came up with 13 or 17-year reproduction cycles, so it may have some affection for prime numbers...
What about the "000" combination? They may be non-viable (boring). They may bear some Down syndrome. Perhaps they existed once as healthy members of the species, but have separated into a new species once the LFSR law kicked in (it prevents them from interbreeding with the others). They may be the phemogian version of non-binary gender.
*And who the hell said the thing with five minutes?
Update: Got in drawing mood today (and calculating):