I just reset and the first time I used the mirror, I got summer. When I came back a little later it was winter. I don't know what changed it: I tried going to the docks and back (as I did before the switch) and going back after gaining a point of reflexes (I had just gained a will point when it changed).
Random and easily disprovable hypothesis:
During your run, it switches to the season associated to the zone you last rested in (Southside - Lo - summer; Docks - David - winter; Metro U - Dr. A - spring(?); Slags - Hel - fall(?))
Update: Hypothesis rejected, but I'm pretty sure I get it now. The seasons change when starting the next part of the main quest and getting access to a new zone. Which season corresponds to what I wrote before.
Started with summer with the southside quest, got winter with the docks quest and now have spring with the university quest.