Adding class to next/prev image link in attachment.php

There two hooks to filter these links: previous_image_link and next_image_link. So we can add these classes per filter:

add_filter( 'previous_image_link', 'wpse_77296_img_link_class' );
add_filter( 'next_image_link',     'wpse_77296_img_link_class' );

/**
 * Add CSS class to image navigation links.
 *
 * @wp-hook previous_image_link
 * @wp-hook next_image_link
 * @param   string $link Complete markup
 * @return  string
 */
function wpse_77296_img_link_class( $link )
{
    $class="next_image_link" === current_filter() ? 'next' : 'prev';

    return str_replace( '<a ', "<a class="$class"", $link );
}