It is many years  since I was current on the factor analysis literature. I have retired and no longer have access to databases of abstracts like DIALOG,  ORBIT, or PsychInfo.  If you have a friend in a university or a government agency they might be able to do a search for you.

Since most clustering grew up around grouping cases (rows in the original data matrix), how is transposing the data matrix and using the same algorithms problematic in clustering variables (columns)?  Just the opposite, one of the oldest methods of clustering cases  was to standardize then transpose the data matrix and factor it.  (this approach was big in the 1960's & 1970's).

I have a gut feeling (not a thought out opinion) that an oblique solution means that you end up with measures that do not have discriminant validity.

SPSS has had many varieties of factor analysis for many years.  It has used 2 kinds of data, 7 kinds of extraction, and 4 kinds of rotation. (56 different "methods"!) Maybe some of those combinations would meet your needs.  [For those of us who use methods that other create,  it sure would be nice if someone were to use this framework and produce a document advising on when to use the options. ]

to get details like algorithms and lit cites go to
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password "guest"
then <catpca> <catreg> <cluster> <discriminant> <factor> <overals> <proximities> <quick cluster> <twostep cluster>

The ANSWERTREE add-on  and new TREE procedure in the base module may also be relevant.

kinds of data: SPSS can work on a correlation matrix or a covariance matrix.  In Psych, the means of variables are usually arbitrary, so correlations are more common. However, much of the development of factoring was from psych and ed.  Perhaps the math psych list would have more current people .
Society for Mathematical Psychology: MPSYCH Listserv

quote from SPSS about the extractions available
Available methods are principal components, unweighted least squares, generalized least squares, maximum likelihood, principal axis factoring, alpha factoring, and image factoring.
end quote.
there are more details in the <help>.

quote from SPSS <help> about the rotations available.   These
end quote.

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Wolfgang M. Hartmann wrote:
Thank you for the nice response,
I kmow that in practice transposing the matrix is a common, but do not think
of it as a very valid approach. (Higher order) Factor analysis with oblique rotation
and restrictions penalizing nonzero loadings would sound good for me. Would
you know of any references for such an approach?

In SPSS all of the few dozen Proximity (similarity measures) can be applied to variables.  (After the data are transformed and transposed)  The  Proximity matrix can then be read into the variety of cluster procedures.  Or the transposed data can be read directly into the CLUSTER, or Quick cluster procedure.  I see no reason (given that you want to cluster variables) that the TWOSTEP cluster could not read a transposed data matrix.

Of course there are all of the varieties of factor analysis which are more commonly used to group variables.  The CATPCA procedure factors categorical variables.

When the variables are used to classify or differentiate a categorical variable, there are procedures like DISCRIMINANT  or the various