you don't have to keep all $arefs after construction.

  DB<112> for $aref ( [1,2,3],[4,5,6],[7,8,9] ) { 
                      $lookup{$aref} = $aref; 
                      $hash{$aref}   = [ reverse @$aref ]; 
                      }
 => ""

  DB<113> \%hash
 => {
  "ARRAY(0xa547e40)" => [9, 8, 7],
  "ARRAY(0xa5c2e68)" => [6, 5, 4],
  "ARRAY(0xa5c3188)" => [3, 2, 1],
}

  DB<114> \%lookup
 => {
  "ARRAY(0xa547e40)" => [7, 8, 9],
  "ARRAY(0xa5c2e68)" => [4, 5, 6],
  "ARRAY(0xa5c3188)" => [1, 2, 3],
}

  DB<115> print "@{$lookup{$_}}\n" for keys %hash
7 8 9
4 5 6
1 2 3
[download]

Hiding all of this behind a tied hash should be feasible, (depending on implementation details of Tie::Hash , IAW when, where and how "stringization" happens )

Anyway I didn't try to find such implementations on CPAN.

One use case could be to implement sets of complex data structures including set operations

updates

I would be more comfortable with a more familiar data structure organization. In addition to familiarity, I think it offers benefits when data elements are added/removed/mutated: as long as data element generation is encapsulated, there are no worries about construction and especially destruction.

>perl -wMstrict -MData::Dump -le
"my @data =
   map data_element($_),
   [1, 2, 3,], [4, 5, 6,], [7, 8, 9,],
   ;
 ;;
 sub data_element {
   return { forward => $_[0], backward => [ reverse @{ $_[0] } ] };
   }
 ;;
 dd \@data;
 $data[2]{forward}[1] == $data[2]{backward}[1] or die 'Oops...';
 ;;
 ;;
 use constant {
   FWD => 0,
   REV => 1,
   };
 ;;
 sub data_element2 {
   return [ @{[]}[FWD, REV] = ($_[0], [ reverse @{$_[0]}]) ];
   }
 ;;
 my @data2 =
   map data_element2($_),
   [ 8, 7, 6, ], [ 5, 4, 3, ], [ 2, 1, 0, ],
   ;
 dd \@data2;
 $data2[0][FWD][1] == $data2[0][REV][1] or die 'Oops2...';
"
[
  { backward => [3, 2, 1], forward => [1, 2, 3] },
  { backward => [6, 5, 4], forward => [4, 5, 6] },
  { backward => [9, 8, 7], forward => [7, 8, 9] },
]

[
  [[8, 7, 6], [6, 7, 8]],
  [[5, 4, 3], [3, 4, 5]],
  [[2, 1, 0], [0, 1, 2]],
]
[download]

I agree about encapsulation, thats why I mentioned Tie::Hash after demonstrating the raw mechanics in a REPL.

If tied or functional or OO or whatever interface... YMMV! :)

Anyway we are again speculating about the OPs real intention, he just showed us enough to think it's pointless for most thinkable motivations.

So pointless that Perl could/should warn about using literal arrays or hashes as keys...

update

tl;dr ...

I just used a reverse in my example to avoid triviality with a minimal amount of code, it was a workaround not a goal.

The general problem is to use complex data-structures as keys, see also Using hashes for set operations... for a similar discussion.

Cheers Rolf

( addicted to the Perl Programming Language)