Subaru Impreza goes in reverse?

I bought (in November 2012) a 2009 Subaru Impreza. On a couple of occasions, when I put the car in Drive, it started going in reverse (in a flat parking lot). Each time, I put the car back in Park, turned the ignition off, started up again and it seemed fine - till yesterday. I was in a parking lot again, tried putting the car in drive and it went in reverse! A few times in a row, till I got so close to the vehicle behind me that I had to call roadside assistance…Any idea what’s going on??

This is a conventional automatic or Subaru’s ‘new’ CVT transmission?
When you shift to Neutral, does it indeed go to Neutral - in other words, you can rev your engine in N and not have the car move? In “N”, can you push the car and it move?
I’m trying it determine whether the linkage somehow has slipped. It would have to have slipped several notches but I guess that could happen.

This is a conventional automatic (I believe the CVT’s came out in the 2010 model or later). When in neutral, I’ve noticed the vehicle move a few times…it moves slowly, as if someone were pushing it. I’ve not tried pushing the vehicle when it was in neutral…

Another, maybe related issue - of late, when I press the accelerator, the engine seems to rev a lot higher (and louder) for a few seconds before the car gains any momentum. It happens more obviously when I am moving from a complete stop. When the vehicle is already in motion, and I try to accelerate, I can still hear the brief higher rev, but it is not as obvious as it is when the car moves from a dead stop.

Does this 2009 have one of those newer sticks, with P-R-N-D and when the ‘sport’ setting, where you can run through the gears, as if it was a manual?
Or is this more a conventional stick, with P-R-N-D-3-2-1?

If the newer stick, I believe that’s mostly electronic. I don’t think it has linkage. My 2009 Legacy does not appear to have it, anyway.
If the conventional, that has actual mechanical linkage.