How to trim adapter sequences from GSE65360 in order to map the reads?
First the Nextera adapters and the custom barcode adapters overlap each other

Nextera 1                                     TCGTCGGCAGCGTCAGATGTGTATAAGAGACAG 
Barcode 1   GATACGGCGACCACCGAGATCTACACTAGATCGCTCGTCGGCAGCGTCAGATGTGTAT 

Nextera 2                                    GTCTCGTGGGCTCGGAGATGTGTATAAGAGACAG
Barcode 2    CAAGCAGAAGACGGCATACGAGATACCCAGCAGTCTCGTGGGCTCGGAGATGTG 


Looking at some read 1s and read 2s you were correct that the adapters appear at the 3' end of the read reverse complimented (The |&gt; points to the adapter in the sequence).  You can see that CTGTCTCTTATACACA would be the ideal sequence to use for the 3' end of the reads.  Since the Nextera adapter and the barcode overlap you only need to use the one that is adjacent to the insert.

SRR1780164_2.fastq: GCCCGAGGGGACGCTGCCCCACGTGCTCCAGG |&gt; CTGTCTCTTATACACATCTGACGCTGCCGACGATGCACGAAGTG
SRR1780164_1.fastq: GGGCTGGACAGCGGCGGGGACGCGGACCTGCAG |&gt; CTGTCTCTTATACACATCTCCGAGCCCACGAGACACAAAGTGA


Of the 64k reads there are ~11k with the Nextera adapter at the 3' end based on a rough sequence grep

On the 5' end of the read there is some Nextera adapter present but it is not reverse complimented and there are only a handful of reads with the adapter present ~250 total between read 1 and read 2.

SRR1780164_2.fastq: TGGGCTCGGAGATGTGTATAAGAGACAG &lt;| CTCCCCCGCCTCCAGCATCCGGGCGAGGTAGTGCATCGACGCGTCCAC


I used this command:

cutadapt --minimum-length=20 -A CTGTCTCTTATACACA -a CTGTCTCTTATACACA -G GATGTGTATAAGAGACAG 
-g GATGTGTATAAGAGACAG -o SRR1780164_1_trimmed.fastq.gz -p SRR1780164_2_trimmed.fastq.gz 
SRR1780164_1.fastq SRR1780164_2.fastq


Which gives a trimming result of 

   === Summary ===

Total read pairs processed:             64,832
  Read 1 with adapter:                  16,721 (25.8%)
  Read 2 with adapter:                  16,996 (26.2%)
Pairs that were too short:                  46 (0.1%)
Pairs written (passing filters):        64,786 (99.9%)


I think this best represents the library construct and the trimming of the adapter sequences