This is an interesting, practical question. But an answer is not so easy to formulate. At least it's not clear whether a solution is practically possible.
Let's assume that the first hypothesis is correct: there are 6 factors which can be combined to form the 24 facets. I put emphasis on this, because these create a 6-dimensional vector space, where you could identify the factors as base vectors.
As far as I understand, you are trying to do two things:
- find questions, which have a high correlation to (the prediction of) the factors itself
- find questions, where you have a proper estimate on the actual value of the factor
Let's assume further, that the "well-crafted" design means that the first part is already solved.
What you hopefully have is some kind of validation data; so that when you have a correlation matrix between the input and the output. If this data is already balanced; e.g. equal numbers among each factors/facets. You can use this to extract the questions that have the smallest variation between members of the same "class", but also a difference between the members of different "classes". Mathematically you are looking for questions which answers have:
- high correlation to the output value/vector component
- small variance
If the data is representative (e.g. has a large number of entries), then you could also possibly extract "corrections" (e.g. matrices). These could be used to correct the deviation in classification for using only subsets of the initial questions. The simple test for this procedure is to verify this with the same data. Mathematically you have an input vector with 240 entries and an output vector of 6 values. Your data processing is formally the matrix, by which you multiply the input vector. You are then looking for the correction matrix, by which you need to alter your "process matrix" to obtain the same output vector.
However, if you don't have such validation data, there is very little you can do that is statistically sound to determine the effect of individual questions.