This is another hypothetical/toy example. I have 3 samples, corresponding to one untreated condition and two treated ones, as encoded in the metadata table below:

    > metadata
       drug
    1  NONE
    2     A
    3     B

Now, I create two deseq datasets, with different designs:

    dds <- DESeq(DESeqDataSetFromMatrix(countData = counts,
                                           colData = metadata,
                                           design = ~ drug))

Finally, I compute results like this:

    results <- list(A = results(dds, contrast = c("drugA", "drugNONE")),
                    B = results(dds, contrast = c("drugB", "drugNONE")))

I'd call the results above "first-order", because they are all differences between the treated conditions and the untreated one.

I am interested, however, in the "second-order" results, namely the "differences between the differences". IOW, how do the effects of drugs A and B (relative to the untreated case) differ.

results(dds, contrast=c("drug","A","none")) gives you A - NONE (on the log scale, i.e. log fold changes)

results(dds, contrast=c("drug","B","none")) gives you B - NONE 

results(dds, contrast=c("drug","B","A")) gives you B - A

You don't need to construct (B - NONE) - (A - NONE) because this is simply (B - A) after some simple algebra.