Methods for analyzing “cross-frequency” tables? I am looking for a statistical method that is suitable for the problem: which goods in my shop are most frequently bought together, and which aren’t?
For my commodities, I can make a table like this where each cell denotes how many times each pair of goods was bought in a single purchase:
             │  Saxophones  Guinea pigs  Astrolabes  Hugs  │  Тotal
Saxophones   │      *           18           1        34   │    61
Guinea pigs  │      –            *           0        10   │    23
Astrolabes   │      –            –           *         4   │     8
Hugs         │      –            –           –         *   │    75

“Total” means the number of purchases that included a given item, regardless of whether it was bought alone or along with something else.
Asterisks are the intersections of an item with itself, that doesn’t make sense in this analysis.
The bottom-left part of the table is filled with dashes, because the numbers in it would duplicate those of the upper-right part. So this is more of a triangle that of a table.
From the table, I can suppose that customers tend to buy saxophones together with guinea pigs (or vice versa?), and they don’t tend to buy guinea pigs together with astrolabes. But I would like to learn some statistical method to analyse these conclusions. In other words, which numbers in the table are significantly larger than can be explained by mere coincidence and which are significantly smaller?
Ideally, I look for a method that can prove or reject two hypotheses for every pair of commodities A and B:


*

*Those who buy A are likely to have need also in B.

*Those who buy A are likely not to have need in B.


A method for analyzing not only pairs, but any sets of n different items, would be helpful as well, for a customer can buy anything from only one item to the package of every item in the shop.
 A: What you have is a special form of contingency table called a square symmetric table (in this case with an irrelevant diagonal.) There exist special methods for such tables, you could maybe want to test a hypothesis of quasi-independence. This is independence away from the diagonal. A book discussing such models (in chapter 8) is Discrete Multivariate Analysis: Theory and Practice. 
For your specific questions 1. and 2. you could calculate (estimate ...) the expected counts under quasi-independence, and then look at ratios of actual count to expected counts. Ratios above 1 would indicate items that attract each other, below 1 items that repel each other. 
R has a package catspec (on CRAN) with such special models. For analyzing not only pairs, but $n$ items, maybe look into market basket analysis. 
With your specific data, I find the ratio of actual counts to expected counts under quasi-independence to be
            Saxophones Guinea pigs Astrolabes Hugs
Saxophones          NA        1.72       0.27 1.00
Guinea pigs         NA          NA       0.00 0.78
Astrolabes          NA          NA         NA 0.89
Hugs                NA          NA         NA   NA

but note that here quasi-independence probabilities are calculated using the counts of total sales, which includes sales of only 1 item. It might be that buyers only buying 1 item are different from buyers buying more, so that can be discussed. 
