An even bigger question is: which measurements actually matter (e.g., once THD, IMD, S/N+D fall below audible levels) and which if any relate to the component's sound?
That depends on the component in question!
If an amplifier or preamp, the distortion spectra at 1 Watt is good. Also distortion spectra a 6dB below full power. Also distortion vs frequency at 1 Watt and -6db of full power. If the distortion overall is -105dB or better then the music can mask the distortion and the amp or preamp will be pretty neutral. Many people think that the distortion can be higher and it will still be inaudible. IME, that isn't true.
Here are some things to help out:
1) the ear assigns a tonality to all forms of distortion. As an example the 'sonic signature' of any amplifier is actually its
distortion signature.
2) the ear uses higher ordered harmonics to sense sound pressure and so is keenly sensitive to their presence; it has about a 130dB range! The higher ordered harmonics as distortion are sensed as 'bright and harsh'.
3) the lower ordered harmonics (2nd and 3rd) can mask the presence of the higher ordered harmonics if the amplitude of the former is sufficient.
If you know these things then the measurements (if they exist as I described) can tell you a lot!
If you see distortion increasing with frequency (above 2-3KHz) that's a sign that the circuit is losing feedback at those higher frequencies. This can happen if the circuit lacks the Gain Bandwidth Product to support the feedback. The increased distortion will be perceived as brightness and harshness. We've been hearing solid state amps with this problem since the inception of transistors. In addition, if the circuit lacks the feedback needed to really clean things up (+30dB at all frequencies) it will sound harsh and bright since otherwise the feedback will cause distortion of its own thru the process of bifurcation at the feedback node, which is non-linear.
Tubes get around this a bit by having a pronounced 2nd or 3rd harmonic which can mask the presence of the higher orders. In this way they can sound smooth in the mids and highs
even though they have more higher ordered harmonics than most solid state amps.
Supporting +30dB of feedback is pretty hard and the semiconductors needed for an amp to do that didn't exist in the 70s or 80s and just barely sometime in the 1990s. Plus the will to deal with this problem for the most part seemed to be absent with many solid state designers. So we have a lot of solid state amps that are harsh and bright as a result.
With class D amps its possible to apply enough feedback so there are class D amps now that lack the harshness of traditional AB amps. Such amps are usually of the self oscillating variety; so much feedback is applied to them that the amp goes into oscillation as soon as its turned on. The oscillation is then used as the switching frequency, killing two birds with one stone. Even then, if the distortion spectra isn't right, the amp can exhibit amusical characteristics. But they can be seen if the amp is tested on the bench as I described.