You obviously can invert 9ths any way you like. The question is - or two questions - do they become different chords, and how do we name them?
I agree this chord is Em9 (or arguably Gmaj13), and if B is in the bass it's Em9/B, or the 2nd inversion. The issue with inverting 9ths is what happens when the 9th is in the bass - that's when it becomes, effectively, a different chord. But it's true I don't know if there's a convention for naming such a "4th inversion".
But this is a fairly common chord shape that guitarists discover, without generally worrying what it's called. Bob Dylan used something very similar as the first chord of the verse of Girl Of The North Country:
-0-
-3-
-0-
-4-
-x-
-0-
That's with E bass, of course, rather than the B on 5th string (so more clearly Em9). Others have D on 5th string as low note, but it's the same principle: take a C (or Am7) and shift it 2 frets up while keeping the open strings. Hence the assumption it's some kind of D (or Bm) chord, because that's what it looks like.
But really it's a kind of hybrid, resisting naming according to the conventional tertian system. It "adds up" to "Em9", if you spell it out, but it may not work or sound like that.
The B-F# 5th on the bottom definitely suggests a "B-root" sound, but then we have to call it something like "Bmadd4b6" - not very elegant.
