Another issue to be aware of is duplicate values. This can be handled by doing a bit more work with the inversion:

my %hash_inverted;
for (keys %hash) {
    my $value = $hash{$_};
    push @{ $hash_inverted{$value} }, $_;
}
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This way each value in the inverted hash will be an array reference, which contains all the keys that value corresponds to.