This solution is non-destructive and uses an array slice rather than push'ing, pop'ing and shift'ing.

use strict;
use warnings;

my @array = qw{ ace big cat dog egg fig gog hog };

my @newArray = @array[ reOrder( scalar @array ) ];

print qq{@newArray\n};

sub reOrder
{
    my $length = shift;
    die qq{Array length not a multiple of 4\n}
       if $length % 4;
    my $low = 0;
    my $high = $length - 1;
    return
       map { $_ % 2 ? ( $high --, $low ++) : ( $low ++, $high --) }
       1 .. $length / 2;
}
[download]

The output.

hog ace big gog fig cat dog egg
[download]

Cheers,

JohnGG

Update: Re-factored subroutine to handle any length of array.

sub reOrder
{
    my $length = shift;
    my $low    = 0;
    my $high   = $length - 1;
    my @slice  =
       map { $_ % 2 ? ( $high --, $low ++) : ( $low ++, $high --) }
       1 .. $length / 2;
    return @slice, $low == $high ? $low : ();
}
[download]