Rotating “up” or “down” — is that like clockwise and counter-clockwise?

When an object is rotating, both the “up” and “down” directions point along the vertical axis. Do they correspond to clockwise and counterclockwise?

Yes. Distinguishing between the two opposite directions of rotation using words alone requires that everyone agree on what to call those two directions. It also requires that everyone have an artifact that they can use to identify which direction is which. When something is spinning about a vertical axis, a carousel or merry-go-round, for example, then physicists name the two possible directions of rotation “up” and “down” and use a right-hand rule to identify which is which. Since most people have a right hand and know which hand it is, the necessary artifact is built-in.

In more common language, the two directions might be called “clockwise as viewed from above” and counter-clockwise as viewed from above”. In this case, the artifact is an old-fashioned analog clock and is probably more of a remembered artifact than one that is in the room with you. Nonetheless, that common naming convention is fine; it’s just wordier than the physicist’s version.