Or is it an American strong ale?
I mean, it could be that, too. The reality is that the difference between and American strong ale, American barleywine, and IIPA (imperial or double, means the same thing) can be very small. Often it just comes down to attenuation and/or final gravity.
For example, Arrogant Bastard is sometimes referred to as an American strong. But it has the hops to put it squarely in the IPA category, and it has the malt character to qualify as a barleywine (if it were perhaps a bit higher abv and FG). So which category does it go in?
The point being, your recipe definitely works as an IPA (at least from the malt bill). You could call it something else if you wanted to, but why? You designed it as an IPA, it would appear, so I'd stick with that description.
As for three styles in one...I disagree. IPA is the style. Imperial is the strength. Rye is a signature ingredient. You're simply applying standard nomenclature. But you could also call it a "IIPA brewed with rye," if that makes it feel less clunky to you.