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Please refer to https://wikisutexas.utexasatlassian.edunet/wiki/display/GSAF/Illumina+-+all+flavors for Illumina library adapter layout.

The top strand, 5' to 3', of a read sequence looks like this.

<P5 capture> <indexRead2> <Read 1 primer> [insert] <Read 2 primer> <indexRead1> <P7 capture>

The -a argument to cutadapt is documented as the "sequence of adapter that was ligated to the 3' end". So we care about the <Read 2 primer> for R1 reads, and the <Read 1 primer> for R2 reads.

The "contaminent" for adapter trimming will be the <Read 2 primer> for R1 reads. There is only one Read 2 primer:

AGATCGGAAGAGCACACGTCTGAACTCCAGTCAC

The "contaminant" for adapter trimming will be the <Read 1 primer> for R2 reads. However, there are three different Read 1 primers, depending on library construction:

TCTACACGTTCAGAGTTCTACAGTCCGACGATCA    # small RNA sequencing primer site
CAGGTTCAGAGTTCTACAGTCCGACGATCA        # "other"
TCTACACTCTTTCCCTACACGACGCTCTTCCGATCT  # TruSeq Read 1 primer site. This is the RC of the R2 adapter

Since R2 reads are the reverse complement of R1 reads, the R2 adapter contaminent will be the RC of the Read 1 primer used.

For ChIP-seq libraries where reads come from both DNA strands, the TruSeq Read 1 primer is always used.
Since it is the RC of the Read 2 primer, its RC is just the Read 1 primer back
Therefore, for ChIP-seq libraries only one cutadapt command is needed:

raries, both R1 and R2 reads}
cutadapt -a GATCGGAAGAGCACACGTCTGAACTCCAGTCAC

For RNAseq libraries, we use the small RNA sequencing primer as the Read 1 primer.
The contaminent is then the RC of this, minus the 1st and last bases:

TCTACACGTTCAGAGTTCTACAGTCCGACGATCA    # R1 primer - small RNA sequencing Read 1 primer site, 5' to 3'
TGATCGTCGGACTGTAGAACTCTGAACGTGTAGA    # R2 adapter contaminent (RC of R1 small RNA sequencing Read 1 primer)