If the same name of notes are re arranged in a different order, (such as F, G#, B, D), this would be called an "INVERSION". The chord still has the same identity or alphabetical name.
If the chord is re written enharmonically (same sound different letters) we have E#,G#,B,D or F,Ab,Cb,Ebb.
Remember, the name of the full diminished 7th chord is derived from the vii degree of some harmonic minor scale.
The diminished 7th chord now belongs to the key of F#, or the Gb harmonic minor mode.
If you think of the note F as being the root of the vii chord, the next note would be Gb for the tonic or eighth degree of the scale. The scale would then be Gb Melodic Minor. The notes of the harmonic minor scale would be Gb, Ab, Bbb, Cb, Db, Eb, F, Gb
If you associate the name of the vii scale degree as being E#, then the next note would be F#, and the start of the harmonic minor scale.
The notes used for the harmonic minor scale would now be F#, G#, A, B, C#, D, E#, F#. The notes in the F# harmonic minor mode (scale) are easier remember than the Gb harmonic minor mode.
If the notes in the arpeggio, E#, G#, B, D, are re arranged, (inverted), the chord would have the same sound, same name, and be in the same key as before. B, D, E#, G#.
