Half-Diminished chords, also referred to as min7b5 chords consist of the Root-b3-b5-b7. Min7b5 chords are also thought of as min7#4 or min7#11.
Let’s take Dmin7b5 (D Half-Diminished) D-F-Ab-C and change the order of notes to F-Ab-C-D. By doing so we now arrive at an Fmin6 chord. So Dmin7b5 has the same notes as Fmin6.
If we change the order of notes to Ab-C-D-F we can technically name this chord Ab6#4 but it’s only an inversion of the Dmin7b5. You would find this inversion notated as Dmin7b5/Ab. If we think of G# as being the enharmonic of Ab, we can conclude that G#-C-D-F is the upper extension of an E7#5b9 without the root.
D-F-Ab-C is the upper extension of a Bb9 without the root.
Since Dmin7b5, Fmin6, E7#5b9 and Bb9 all have the same notes try substituting one voicing or chord form for the other. Try playing the following forms for each of the above chords we just analyzed.