In search of differences of non-mutually exclusive k categories over g groups

I conceptualize this problem to be similar to a 2x4 test of independence with a chi squared test statistic, but I recognize the problem is different.

Suppose we have 2 groups with 20 participants in each group. The groups are mutually exclusive. For any given participant they have some characteristic that is not mutually exclusive in that the levels are A, B, C, and D may be present in a participant in any combination. Participant 1 may have A, C, and D true while participant 2 may only have C, for example. To make the level more concrete, it may be something like UP&LEFT UP&RIGHT DOWN&LEFT DOWN&RIGHT. The person may have at least 1 of these true for their characteristic and up to all 4 simultaneously. The issue originally was that up and down can occur in the same participant as can left and right, so the other presentation is a dummy for each 1 if yes for up, 1 if yes for down,... so on.

The idea is to see if there is some difference in the proportion or frequencies of these levels across the two groups. I'm just not sure it should be analyzed by the traditional 2xK contingency table since each participant can contribute to more than one cell.

Thoughts are appreciated.
Hmm, without a digestible example it is a little difficult to process the context. So you don't have an outcome of interest, you just want to compare the frequencies? Or do the two groups represent the outcome of interest? Non-mutual exclusivity is always an issue in describing all of the combinations, things get sparse for sure.

What is the purpose exactly?