Duh! Okay, I think that I fixed part of that problem. In the products_categories table, I have only two fields, the product_id and category_id. I was simply using the wrong one 🙁
So I got this to work like a charm, but without any the IN clause:
$sql = "SELECT id, title, price, author, picture, description FROM products, products_categories WHERE ((products.picture !='') AND (products.id = products_categories.product_id)) ORDER BY rand()";
but with the IN clause, I still get no result set. Also, running it from the command line produces an error which it doesn't define:
$sql = "SELECT id, title, price, author, picture, description FROM products, products_categories WHERE ((products.picture !='') AND (products_categories.category_id IN (405,416,418,420,430,440,510,580,605,730,792)) AND (products.id = products_categories.product_id)) ORDER BY rand()";
Thanks so much. I really appreciate the help.