I'm having trouble making sense of the 2 traces. I'd like to see them both on the screen at the same time. It looks like you have a 4-channel scope. How about one channel primary voltage, one channel secondary voltage, and as I think oldway is suggesting, one channel with a clamp-on current probe, showing primary current. At that secondary voltage, I'm assuming it's not an autotransformer so the primary and secondary are isolated, right?
Also, what's going on with that porch -- after the positive going half cycle gets to zero, it keeps going and levels off negative, before the negative half cycle starts. This is part of the assymetry that oldway is talking about, in addition to the different peak values. Have you tried a different triac to see if you can get something more symmetric?
Something else I'd be interested in seeing, just to get a feel for what's happening: Gate your triac for nearly full 360 degree conduction, and verify that you're getting something normal looking on the secondary. Then slowly reduce the conduction time and watch to see where the funny stuff starts. If you can see that plus the primary current as this is happening, I think it would help you understand what's happening.
You're probably aware that power transformers are generally not designed to take an instantaneous voltage step like a triac turning on. They prefer to see the time-varying sine wave. You can't really change the primary current instantaneously, so that's why traces including primary current would be interesting to see. I'm not sure what your end goal is, but you can tell us about it if you like. But maybe there's another way to get to your end goal.