How to reactivate a closed palette?

You can click the X in a palette to close it, but what is the way to reactivate the closed palette?

The same user action that activated it in the first place should show the palette again. If that was switching to an app, you may need to switch to another app and then back.

The triggering event is the popping up of a dialogue box:

I would expect that when I click the X in the title bar of the palette, this not only makes the palette disappear but also would turn the check mark into a cross, indicating that the palette group has been disabled. Reactivating would than be as simple as clicking the cross to turn it into a check mark.

No, activation and enabled states are two different things.

The enabled state (signified by the large tick/cross in the macro/macro group) is a semi-permanent state.

The activation state (configured by the settings in the macro group) is a transient state, changing as things on your Mac change (like the front application or window title).

In this case, clicking the close box on the palette should just deactivate (not disable!) the macro group, which will then re-activate the next time the macro group should be activated (although the activation state code for macro groups is horrendously complex, so it would not overly surprise me if the behaviour was less that clear and obvious).