use strict;
use warnings;
use Data::Dump::Streamer;

my %hash = (
    doubleCheeseBurger => 'burger',
    cheeseBurger => 'burger',
    cheeseSandwich => 'sandwich',
    tomatoSandwich => 'sandwich',
    );
my %rHash;

push @{$rHash{$hash{$_}}}, $_ for keys %hash;

Dump (\%hash, \%rHash);