Swaybar for regular driving?

[dogsridewith]

Originally Posted by tmontague

....both rear wheels are attached through a solid h style axle.

The cross-the-car part of this piece isn't "solid". It is torsionally soft to do what a sway bar does. You might say it replaces separate trailing arms, two lateral links and a sway bar with a single piece. That was VW's invention, and why Toyota uses it.
This is not off-topic. It will be fine for OP to know more about his suspension as he modifies it...particularly that it already includes some roll control.
(In a very crude search of the Yaris rear suspension, I'm finding the terms "torsion beam rear suspension," and "semi-independent rear suspension." The main thing is to understand how it works.)