Given a set of purchase records, I would like to find out which products are often bought together. Is logistic PCA a sensible method to accomplish that? Are there any clustering methods for that purpose? I would appreciate any pointers to methods that can help with that.
This is often called „market basket analysis“ or association mining. The idea is to find co-occurance of purchased products. There are many good tutorials/solutions/packages out there, so it should be easy for you to get a good start.
Determining pairs of products which are frequently bought together is different from clustering. For example, if products A and B are often bought together and products B and C as well, it's possible that A and C would belong to the same cluster even though A and C are not frequently bought together. There's no transitivity, so in general it's not a very good case for clustering, as opposed to "similar products" for instance.
If the goal is actually "frequently bought together", it can be done directly from a matrix of cooccurrences using joint frequency directly, or using a measure of statistical association such as Pointwise Mutual Information. The latter takes into account the conditional probabilities, i.e. if product A is often bought with product B but product B is also frequently bought with many other products, it will have a lower PMI than some pair of products which is less frequent but more "exclusive".