This is one of the very few good uses for prototypes that I know of. (I forget who pointed this one out to me, but it was on perlmonks.)

Just have your sort functions declared with a prototype of ($$) and it will pass data in through @_ and never use $a or $b. (Avoiding confusion about what package $a and $b are to be found in.) That is write:

sub mycmp ($$) {
  my ($first, $second) = @_;
  return 1 if $first ne ( mysort($first, $second) );
  return -1 if $first eq ( mysort($second, $first) );
  return 0;
}
# and elsewhere
  ... sort {&Whatever::Package::mycmp} map ...
# or put in in a variable
  my $cmp = \&Whatever::Package::mycmp;
  ... sort $cmp map ...
# of if you don't mind a speed hit and don't trust the user
  my $cmp = sub ($$) {Whatever::Package::mycmp(@_)};
  ... sort $cmp map ...
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(Should work in 5.6 on up.)

Incidentally coding a cmp function in terms of a sort function strikes me as both obscure and inefficient.