I am proposing here a variant of this problem.
Let $p$ and $q$ be distinct odd primes. Is it true that there exists a permutation $\sigma$ of $\{1,\ldots,2pq\}\times \{1,2\}$ such that $$ \{\sigma(1,x),\ldots,(2pq)\sigma(2pq,x)\}=\{1,\ldots,2pq\} $$ modulo $2pq$ for each $x \in \{1,2\}$?
[I think the answer is affirmative, as in the other case]
The question is related to this one. In particular, it follows that the answer is affirmative if $p=3$ and $q=5$.
