Coming across the six modal verbs in the German language, I've put them in correspondence (1:1) with English ones (or similar English verbs used in conjunction with main verbs):

1a. mußen --> must

1b. sollen --> should/has to

2a. wollen --> want

2b. mögen --> like/would like

3a. dürfen --> may/be allowed to

3b. können --> can

In this way category 1 expresses a duty (mandatory or advised); category 2 expresses willingness; category 3 expresses the potential to do something.

I'd like to know whether this correspondence, especially within the same category, is correct, or if there are some details I'm missing.