I'm not sure what you're asking but
you can't hear a phase shift without a "reference". If there is a phase-difference between the left & right channels you might hear the
results of the phase shift when the soundwaves combine in the air.
A few examples -
The crossover in a speaker often introduces phase shifts. The woofer tends to have a phase-lag and the tweeter a phase-lead (at the crossover frequency). In a 2-way system the woofer might have a 90 degree lag and the tweeter a 90 degree lead at the crossover frequency. That puts the soundwaves out-of-phase and they will cancel, creating a dip in frequency response. The solution is to flip the polarity of the tweeter which makes the drivers in-phase at the crossover frequency and the tweeter 180 degrees from where it would otherwise be at higher frequencies. In a 3-way speaker the midrange connections would be reversed.
If you reverse the connections to one stereo speaker (making a 180 degree phase inversion) you'll get a weird "spacey" sound and the bass soundwaves will mostly cancel, giving you weak bass. If you invert both connections it will sound normal.
If you flip the polarity of one headphone you (probably) won't hear the effects because the soundwaves don't combine.
A lot of equipment flips/inverts the phase. There is no "rule" that an amplifier can't invert the phase and sometimes it's more convenient to build an inverting amplifier. Equipment in the studio might invert the phase so some recordings (or some instruments or voices on recordings) are inverted.
The process of cutting and playing a vinyl record (including the RIAA record/playback EQ) introduces different phase shifts at different frequencies.
Delay creates phase shifts. The distance from the speaker and the speed of sound means that high frequencies are "shifted" by thousands of degrees at your ears. The bass is probably "shifted" only a few degrees (with the short distances in your living room).
Because of the distance between your ears, the phase is different in each ear. And if you play a 10kHz test tone (about 1.3 inch wavelength) from both speakers you'll hear it get stronger or weaker as you move your head slightly and the waves from both speakers, and the reflected waves, go in-and-out of phase with each other. The same thing happens with regular music but we don't notice it because music contains many frequencies, and we are used to it.
Here are a couple of posts by Amirm -
Does Phase Distortion/Shift Matter In Audio (no)
Understanding Phase