Etydhv

If I had a recording of some music, I could play it backwards. This is pretty simple.

If I had some music recorded, could I flip it around a note like in this image? If possible, how does that work?

No, there is no single-pass audio process that can invert notes or melodies. Something like Celemony Melodyn Studio could get you started by analyzing all the notes, and then maybe you could embark on a lengthy MIDI inversion process and then try to use the inverted MIDI to re-arrange the audio in Melodyn. That could take a long time.

Theoretically, this could be done by adapting the techniques used in a vocoder.

A vocoder works by measuring the amplitude of different frequency bands in a modulator signal (e.g. vocals), and then using that information to filter a carrier signal (e.g. a synthesizer sound), giving it a similar frequency spectrum as the modulator signal.

If you inverted the connections, you could control the low frequencies in the carrier signal with the high frequencies in the modulator signal, and vice versa. In fact, patching the connections in a non-linear way was a feature of some analog vocoders, like the EMS Vocoder 5000 from 1975 and the Moog 16-Channel Vocoder from 1979.

^{Moog 16-Channel Vocoder (1979) with frequency band switching patch bay (image from vintagesynth.com)}

To achieve completely frequency-mirrored audio, you'd have to use white noise (which contains all frequencies) as the carrier signal, and you would need to have many more frequency bands than the dozen or so of a classic vocoder. Software vocoders like Reason's BV512 can have hundreds of bands, but you'd probably need thousands to achieve anything "realistic".

Even then, the result would not be music with mirrored note pitches. Every sound would have its component harmonics mirrored, and the result would be unrecognizable and unmusical.

This smacks of negative harmony, and I highly recommend you read that post. It's very informative, and one of the processes involved in that discussion is essentially inverting a song about an arbitrary axis, which seems to be what you want. However, though this is possible, I don't know of any software to do it for you, and generally this is a (rather rare) conceptual device used by musicians and applied to music, not just an algorithm that gets applied to existing sheet music to get a new result.