relativity - Wormholes, time travel, and time dilation

I believe I understand the "wormhole time machine" now, but still have one question regarding what happens if you enter the stationary mouth.

A wormhole is created and (of course) has two mouths, $x$ and $y$.

$x$ remains stationary on Earth while $y$ is sent away at relativistic speeds (.99c) for (example) five years as measured on Earth (observer at the stationary mouth $x$).

The spaceship with mouth $y$ returns to Earth. Mouth $y$ and any occupants of the spaceship have obviously aged less than five years, though observers on Earth and wormhole mouth $x$ have aged 5.

What now happens if someone enters into wormhole mouth $y$?

As explained below by Ben Crowell, you would exit mouth $x$ at a time in the past. This happens because the time (as measured by clocks, if you wish) inside the mouths of the wormholes remains syncronized. So, when you enter $y$ (the younger wormhole mouth), it takes you to the mouth of $x$ when it was the same age. You end up in the past.

What happens if someone enters into wormhole mouth $x$?

I can't find a definite answer on this, so I'll try to work through it. I think you would emerge from mouth $y$ - the one in the present time (5 years old). Because the time inside remains synchronized between the wormholes regardless of the outside age, then entering $y$ takes us to $x$ in the past (sometime between <5 and 5 years ago); however, entering $x$ does not take us to the future because mouth $y$ (outside) has not yet aged beyond 5 years since we started our experiment.

Can someone confirm if my reasoning is correct or not?