I figured out what I was doing wrong. A simple beginners mistake.
Array_reverse was working properly, but I wasn’t then reassigning the reversed array back to the $home_shows WP_Query, hence not seeing any change.
Here is the answer, and my revised code.
<?php
$args = array(
'post_type' => 'show',
'posts_per_page' => 5,
'order' => 'DESC',
);
$home_shows = new WP_Query($args);
//reverse the order of the posts, latest last
$array_rev = array_reverse($home_shows->posts);
//reassign the reversed posts array to the $home_shows object
$home_shows->posts = $array_rev;
?>
<?php $captions = array(); ?>
<?php if ( $home_shows->have_posts() ) : ?>
<?php while ( $home_shows->have_posts() ) : $home_shows->the_post(); ?>
Thanks for the replies, glad I figured this one out.