First of all $a and $b are special in Perl so it is a good idea to avoid using them, even for throw-away variables.

That aside, this method of swapping data is unreliable. Witness:

my $x = ["hello"];
my $y = ["world"];

print "We have \$x=$x and \$y=$y.\n";
print "The first element of \$x is '$x->[0]'\n";
print "The first element of \$y is '$y->[0]'\n";
print "Now swapping...\n";

$x = $x ^ $y;
$y = $x ^ $y;
$x = $x ^ $y;

print "We have \$x=$x and \$y=$y.\n";
print "The first element of \$x is '$x->[0]'\n";
print "The first element of \$y is '$y->[0]'\n";
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What happened, of course, is that you stringified the object and so lost the reference.

Now if you want a sneaky way to swap variables, the following takes a list of variables as arguments and rotates them. The first goes to the end, the rest move forward one. With 2 variables this just swaps.

my $x = ["hello"];
my $y = ["world"];

print "We have \$x=$x and \$y=$y.\n";
print "The first element of \$x is '$x->[0]'\n";
print "The first element of \$y is '$y->[0]'\n";
print "Now swapping...\n";

rotate($x, $y);

print "We have \$x=$x and \$y=$y.\n";
print "The first element of \$x is '$x->[0]'\n";
print "The first element of \$y is '$y->[0]'\n";

# And here is the magic bit
sub rotate { @_[0..$#_] = @_[1..$#_, 0]; }

# BTW here is something that won't work.  Lists are not arrays! :-)
sub wont_rotate { @_ = @_[1..$#_, 0]; }
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