A little late to the party, but I didn't see any other posts that leveraged the cyclical nature of Z₁₀, so seemed worth putting in my two cents. Should be pretty efficient as well, given the once-through approach.

#!/usr/bin/perl -w
use strict;

while (<DATA>) {
    chomp;
    my $max = 0;
    my $count = 1;
    my $direction = 0;
    my $last = substr $_, 0, 1;
    for my $this (split //) {
        local $_ = $this - $last;
        ++$count and next if $direction and $_ % 10 == $direction;
        $count = 2 if $direction = abs==1 || abs==9 ? $_ % 10 : 0;
    } continue {
        $max = $max > $count ? $max : $count;
        $last = $this;
    }
    print "$_\t=> $max\n";
}

__DATA__
82665409266027476709324472
2468
2345678
78901
78909
32109
32101
909
09090
[download]

outputs

82665409266027476709324472    => 3
2468    => 1
2345678    => 7
78901    => 5
78909    => 4
32109    => 5
32101    => 4
909    => 2
09090    => 2
[download]

Update: Ambrus caught a mistake in my $direction assignment for the case of 909. Corrected bugs (changed $this <=> $last to $_ % 10) and added cases to test set.