My first thought was:

our $flipflop = !1;
our %class = ( $flipflop => 'even', !$flipflop => 'odd' ); 

sub getclass {
    $class{ $flipflop = !$flipflop };
}
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The '!' in the initialisation values are to avoid reliance on specific truth values so they can be used safely as hash keys for any given Perl implementation.

update: in practice I'd use an instance variable instead of a class variable for $flipflop, but before that I'd have to do OO makeover of the OP which I avoided doing.