First, nomenclature: it is gravitational waves, not gravity waves. Gravity waves are things like waves on a water surface.
Second, you cannot really learn much about gravitational waves by watching simulations like this, which will always be some lower dimensional representation of the full simulation.
It is not clear to me what you would mean by ”opposite metric”. The metric of a spacetime is by construction Lorentzian everywhere.
In particular coordinate systems, gravitational waves may be described as
space expanding and contracting (not spacetime, which is what it is - time is a part of it so it does not actually
do anything). However, that is a coordinate dependent interpretation requiring you to make particular choices of coordinates and assign them the space and time labels.
desertsoldier39 said:
When spacetime contracts at the leading edge of the wave and then expands, does the expansion revert back to normal spacetime flatness or is there a momentary expansion beyond that metric?
This completely depends on the shape of the wave - as for any wave, gravitational waves are governed by the wave equation.