"An important property of an impulse, not intuitively obvious, is that it if you break it up into individual sine waves you find that it contains all frequencies at the same amplitude. Strange but true."
"So my understanding is they are two different representations of the same thing and can be transformed back and forth."
Not quite. You can get the frequency response from the impulse response, but not the other way around. This is because the equal-amplitude sine waves in the (perfect) impulse have another property: they all have a maximum at time zero (the time of the impulse). But a flawed speaker (and they're all flawed) will not put all the peaks at time zero, but have the peaks at slightly different times. This is called "phase response". If you know both the "frequency respsone" (the amplitude of each sine wave) and the "phase response" (the offset of each sine-wave maximum from time zero), then you have enough info to reconstruct the impulse ressponse (what the speaker produces given a perfect input impulse).
Incidentally, a perfect impulse is mathematically a "Dirac delta function", invented by physicist Paul Dirac:
https://en.wikipedia.org/wiki/Dirac_delta_function. I'm pretty sure this is why the Dirac room correction system was given that name.