Possibly this question depends on location. I'm most interested in the US convention.
I have read here a convincing claim about sink faucet handles: when there are separate hot and cold handles, cross handles (knobs) should always turn on counter-clockwise, but lever handles which are off pointing outward should turn on by pulling them toward you: that's counter-clockwise on for the left, hot handle, but clockwise on for the right, cold handle.
What about a double lever handle for a shower, where the two handles, which are wall-mounted, are pointing outward and might turn either up or down?
Here, you are generally standing looking down at the handles. Is down on, which would be counter-clockwise for the left/hot handle and clockwise for the right/cold handle, matching the sink configuration? Or is up on, which has the same feeling of "pull toward you" but is actually opposite in terms of the direction of rotation?
Best Answer
It seems to vary by manufacturer/model/style and perhaps personal preference.
The shower knobs in one of my bathrooms were installed 2-3 years ago and are, by default, "off" when horizontal and "on" when vertical. Cold turns on CW and Hot turns on CCW.
Below is an excerpt from the installation instructions for the faucet set that shows this as the way the handles were designed. This does not strike me as being unusual thinking back to some of the other places I've lived and so forth.
Interestingly, the faucet set is prominently described by the manufacturer as being "ADA Compliant" (compliant with the American Disabilities Act) so it would seem that the on/off direction isn't critical for that designation. Incidentally, ADA compliance isn't required for private residences.
It makes good sense that vertical be "off" and horizontal be "on" for the reasons mentioned in other answers, as well as not having your shower/bathtub water come on if something is dropped on the handles. (I've done it. Rather eye-opening surprise...)
Here's a link to the whole installation manual in case someone wants it.