Thanks, David. I will save that info for future problems.

Here's how I finally resolved it. When my cursor hovered over Pegasus in the task bar, a small grey Pegasus box appeared. It was THAT box that I was able to right-click on, and finally see the move/minimize/maximize. When I maximized, there it was. It took a long while of I don't know what, really, before I was able to stretch that damn little grey minimized box out to its normal size. So it wasn't off screen, it just got really tiny. This same thing has sometimes happened to me while inside pegasus, where I've suddenly 'lost' the email I was typing. Eventually I find it in the bottom left corner, looking like it's been minimized but really needs stretching up and out (forgive my terminology - I'm sure it's not quite right). 

When I open Pegasus and sign in under my usual name, it acts the way it's supposed to, with that little counter for the new mail folder, but the Pegasus application screen is nowhere. I've tried cascading the windows, it's not there. I can see it when I alt-tab, and I make it the active window, I can see it on the task bar but have no 'move' option. I can see it's working - my NEW MAIL flag shows that there is new mail. It's just gone.

I have more than one email account and it works perfectly fine on the other one which I hardly ever use.  

The window might be off-screen. Try this