Weird hysteresis behavior
Your opamp stays clipped on one rail, then swings into clipping to the other rail. Most likely what you're seeing is its clipping behavior.
Ideal clipping behavior would be like this:

A real amplifier will never be this pretty. Here's an example of sticky clipping:

So it takes a while for any opamp to come out of clipping and return to linear behavior. On your scope shot, this looks like hysteresis, but I bet if you change the frequency of the sawtooth, you will notice what you think is &quot;hysteresis voltage&quot; will change, because it is actually a delay between the input crossing zero and the output reacting.
The cause of this delay is pretty simple, but difficult to eliminate. Here's a newbie CMOS opamp.

When it clips, FET T5 will be either fully on, or fully off, so its Vgs will be either 1) way above its normal operating point, or 2) close to zero. To come out of clipping Vgs has to be brought back to its operating point, but this takes a while because the FET gate is a capacitor and the only current available to charge or discharge it is the small output current delivered by the input stage.