Even males have natural hormone cycles, so I don't think it's impossible. Though it does appear unlikely. It's not like your body produces or stores the hormone; you yourself administer it at a fixed dose. Though it might be something else that is cycling. For example, the small amounts of E, T and P that your adrenal glands make. Or FSH/LH. Or thyroidian hormones. Or anything. But whatever it is, I doubt it's linked to HRt itself. Actually, if it does happen, it's probably something that has happened all your life. Maybe before it was just masked by the heavily distorting effects T has on character. You couldn't feel more excited, since T is like freaking Viagra 24/7. You couldn't feel more upset, since T made you feel pissed most of the time anyway. Or something like that.
There may be a psychological factor: your brain tells your body it should be cycling, so it looks like it's cycling. Or you might be just more attracted to your girlfriend when she has her period. Pheromones or something.