DESeq2: 5 Conditions (B vs C) vs (D vs B)

Consider a setup with 5 Conditions A-E, where E is the untreated condition and hence the reference label.

I want to know which genes are upregulated in B vs C but are downregulated in D vs B. All samples should be compared to E (the untreated condition).

coldata
        condition
    A_1 "A"      
    A_2 "A"      
    A_3 "A"      
    B_1 "B"      
    B_2 "B"      
    B_3 "B"      
    C_1 "C"      
    C_2 "C"      
    C_3 "C"      
    D_1 "D"      
    D_2 "D"      
    D_3 "D"      
    E_1 "E"      
    E_2 "E"      
    E_3 "E"

    dds_f <- DESeqDataSetFromMatrix(countData = data_f, 
                                    colData = coldata,
                                    design = ~ condition)

    dds_f$condition <- relevel(dds_f$condition, ref = "E")
    dds_f <- DESeq(dds_f)
    resultsNames(dds_f)

    [1] "Intercept"        "condition_A_vs_E" "condition_B_vs_E" "condition_C_vs_E" "condition_D_vs_E"

    upd <- results(dds_f, listValues = c(1, -1/2), 
                           contrast = list(c("condition_B_vs_E"), c("condition_C_vs_E", "condition_D_vs_E")))
    rn <- rownames(upd[!is.na(upd$padj) & upd$padj <= 0.05 & upd$log2FoldChange >= 1, ])

My questions are:

1.) I want to find genes which are downregulated in (condition_D_vs_E vs condition_B_vs_E) but upregulated in (condition_B_vs_E vs condition_C_vs_E)

2.) Is this the right comparison (condition_B_vs_E vs condition_C_vs_E) vs (condition_D_vs_E vs condition_B_vs_E)?

3.) Does the contrast I choose answer this question?

*2.) or does one has to interpret this contrast like this: On average condition_B_vs_E is smaller or higher than condition_C_vs_E and condition_D_vs_E? Meaning that it could happen that this could result in a positive log2 fold change even though the gene expression levels of condition_D_vs_E are actually higher than in condition_B_vs_E? Which is because of the average of condition_D_vs_E with condition_C_vs_E, which in this case contains very low expressed genes.

This comes up a few times on the support forum. Comparing X vs Y, with both compared to Z, is equivalent to X vs Y alone. The algebra on the log scale is:

(X - Z) - (Y - Z) = X - Y

DESeq2 can form the contrasts for you if you just do:

results(dds, contrast=c("condition","D","B"))

You don't have to bother with the list here.

If you want downregulated in D vs B but upregulated in B vs C, you should build two results tables, the first one for D vs B and the second one for B vs C, and then just take the intersection. You can specify altHypothesis="greaterAbs" or "lessAbs" to perform one sided testing.