How to start animation with different cycle offsets (2d)?

Using Unity 5, mecanim animation on sprites, with state machines and transitions and all that good stuff.

I have a multilegged character, and I'm trying to build a prefab for a single leg that I can use four times. But I want the animations for each leg to start at a different offset.

I have the state machine, transitions, etc. all working fine. But each of the four legs animates at the same time. What I really want to do is start the back leg animations halfway through the animation cycle.

I found Cycle Offset -- but it's a setting on the animation, and I don't know how or where is the right place to set it in code. I thought it would be something I could set in a state behavior's OnStateEnter, but I can't find a method call that works.

I found that you could set it using animator.Play(), but... that doesn't seem compatible with the state machine + transition approach.

Edit: I tried using Play() in OnStateEnter, but it looks like that triggers the state again as it gets into an endless loop of starting the animation over and over.

Ideas? Suggestions?