You're close, but not close enough. Let's look at what you're doing:

Your use constant initializes the constant MONTH to a reference to a hash. You want the "inverse" of that hash, which actually happens to exist, as we know that the numbers are unique.

Normally, one can get the inverse of a hash (provided that it exists) by using :

my %forward = (
  a => 1,
  b => 2,
);
for (sort keys %forward) {
  print $_, ":", $forward{$_}, "\n";
};

my %reverse = reverse %forward;
for (sort keys %reverse) {
  print $_, ":", $reverse{$_}, "\n";
};
[download]

Your use of the constant complicates matters a bit due to the inner workings of use constant and how perl interprets things. The following code works:

use strict;
use warnings;
use constant MONTH => { jan => 1,
            feb => 2,
            mar => 3,
            apr => 4,
            may => 5,
            jun => 6,
            jui => 7,
            aug => 8,
            sep => 9,
            oct => 10,
            nov => 11,
            dec => 12 };

my %reverse_month = reverse %{ +MONTH };

print $_, ":", $reverse_month{$_},"\n"
  for sort { $a <=> $b} keys %reverse_month;
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Note that I had to specifically use +MONTH, where one would like to see MONTH, because perl thinks that %{MONTH} refers to the (global) variable %MONTH, while it knows that +MONTH refers to the subroutine MONTH, which use constant defines for you.