I independantly came up with the same thing! but since mine doesn't assume 1,1 exists and since mine detects unconnected polygons, I'll post it.

# -- input --

my @r = ( 
  [ 5, 3 => 5, 5 ],
  [ 1, 1 => 2, 1 ],
  [ 2, 1 => 2, 3 ],
  [ 5, 5 => 1, 5 ],
  [ 1, 5 => 1, 1 ],
  [ 2, 3 => 5, 3 ],
);


# -- work --

%lookup = map { join(',', $_->[0], $_->[1]) => join(',', $_->[2], $_->
+[3]) } @r;

$start = $next = (sort(keys(%lookup)))[0];
#$start = $next = (keys(%lookup))[0];
do
{
   push(@x, [ split(/\,/, $next) ]);
   $next = $lookup{$next};
   die("broken!\n") unless defined($next);
}
while ($next ne $start);


# -- output --

printf("[ %d, %d ]\n", @$_) foreach (@x);



__END__
output:
[ 1, 1 ]
[ 2, 1 ]
[ 2, 3 ]
[ 5, 3 ]
[ 5, 5 ]
[ 1, 5 ]

time complexity:
O(n)  (without the unneccesary sort)
[download]

One thing it doesn't do is detects dots used more than once, so let's fix that:

%lookup = map { join(',', $_->[0], $_->[1]) => join(',', $_->[2], $_->
+[3]) } @r;

$start = $next = (sort(keys(%lookup)))[0];
#$start = $next = (keys(%lookup))[0];
do
{
   $current = $next;
   push(@x, [ split(/,/, $current) ]);
   $next = $lookup{$current};
   delete($lookup{$current});
   die("broken or repeat!\n") unless defined($next);
}
while ($next ne $start);
[download]