CW signals are not “transmitted on the upper sideband”, nor the lower one. A CW signal is approximately at a single frequency (with only the additional bandwidth required to allow the key-up and key-down transitions).
However, the standard method of receiving a CW signal is identical in structure to a single-sideband receiver. The local oscillator (LO) of the receiver is set to a frequency not quite equal to the CW signal, and the difference between the incoming CW and the LO is the audio tone you hear. Let's say for concreteness that the difference is 800 Hz.
If that frequency shifting were all that was going on, then you would hear many CW signals (from both sidebands) at different pitches, so there is a very narrow filter to select just the one signal. That filter will select either the frequency +800 Hz from the LO (upper sideband) or the one −800 Hz from the LO (lower sideband).
For convenience, a modern transceiver does not display the LO frequency (as it does in single-sideband mode) but rather the LO frequency offset by 800 Hz, so that you know the actual frequency of the signal.
If you switch from upper-sideband reception to lower-sideband reception, then you (or your receiver's CPU, more likely) should simultaneously change the LO frequency to be 1600 Hz (double the offset) higher, so that (if you were perfectly tuned to start with) you'll hear the same signal at the same pitch as before.
The difference this makes is that signals not exactly at the 800 Hz offset frequency will be flipped around that midpoint: an unwanted signal within the filter passband that was at (relative) 1200 Hz will now be at 400 Hz, and you might find that lower-pitched sound less troublesome. Or, if your receiver's filters are not perfectly symmetrical for this purpose, the unwanted signal could be actually attenuated.
If you have a receiver already, listen to CW and SSB signals using both CW and SSB modes, tune around, and try to get a feel for what's going on. CW mode is just SSB mode with a narrower filter and the offset display.
(Note that when you switch between CW and SSB mode, your receiver might preserve the actual LO frequency (thus changing the displayed frequency), or preserve the displayed frequency (thus changing the LO frequency and the pitch of what you hear).