skeleton.UpdateWorldTransform checks the skeleton.flipX field to tell where the skeleton is facing.
If you want it to happen on the same frame, you have to set skeleton.FlipX before skeletonAnimation.Update calls skeleton.UpdateWorldTransform.
It's up to you how you want/need to do that.
You can manually set the script execution order. You can use event callbacks. You could also call skeleton.UpdateWorldTransform again (performance cost scales with skeleton complexity and number of skeletons, obviously).