My answers should be comments, but I am, unfortunately, more verbose. For a single handle, consider making the double cover of a Klein bottle as follows. Take an immersed Mobius band which is half the bottle, and make it out of paper. Now double the paper. Of course you can't do this physically since the paper has to pass through itself. But if you make a standard Mobius band and double it, you'll see that you have constructed an orientation double cover. You can double the surface to get a torus double covering the Klein bottle. Let it pass through itself.

So if you see how to regularly homotope an embedded torus to the double cover of the Klein bottle, you are done with this example. That is the gist of Ian's and Ryan's comments.

I was told at sometime that Shapiro's idea is to do the same thing with Boy's surface. You can pass the orientation double cover through itself. The problem is to find a regular homotopy from an embedded sphere to Giller's surface.

No one that I know of (including me) has drawn a reasonable diagram of Giller's surface. It can be done using a movie of Boy's surface, doubling the loops, and tracing the critical points and crossings through the movie. The techniques for making such a diagram are found in my book . As I said therein, the problems with outside-in and Morin's construction is that the symmetric middle pictures are highly singular.

Such surfaces (including and especially outside-in) and regular homotopies *should* be made out of some translucent materials --- colored glass. They would be exquisite to look at.