I'd like to create a home page, where I would like to insert the section with portfolio items in between the content of the page. If I choose Portfolio Mixed template for the page, the portfolio is either shown in the beginning or in the end. Is there a way that I could include those items in the middle?

If anyone needs it, I solved by adding portfolio wrap from portfolio mixed to another .php file, set sanitize_html_class to custom values manually (unfortunately it's not dynamic, but easy to change in code if necessary) and created a new shortcode in shortcodes.php that calls template parts:

function template_part( $atts, $content = null ){
    $tp_atts = shortcode_atts(array(
       'path' =>  null,
    ), $atts);
    ob_start();
    get_template_part($tp_atts['path']);
    $ret = ob_get_contents();
    ob_end_clean();
    return $ret;
 }
 add_shortcode('template_part', 'template_part');

Then, finally called that template with the shortcode:

[template_part path="your_custom_file_name"]