The cat gagged and twitched but didn't vomit.
The reasons why a cat may gag and twitch but not vomit are as follows:
1. Cats may have hairballs in their stomach and intestines. This is because cats have a habit of carrying hair for a long time, and their tongues have many barbs. Hair can easily enter the digestive tract during the process of carrying hair. Hair can accumulate in the stomach and form hairballs. If there are too many hairballs, it can cause cats to gag.
2. If a cat ingests foreign objects such as plastic rope, it may also cause the cat to gag without vomiting. 
Treatment for hairballs:
To deal with hairballs, the first step is to develop the habit of grooming your cat regularly. You should brush your cat's fur regularly and clean up any loose fur promptly. Otherwise, your cat may easily ingest the fur. Regularly brushing your cat's fur also helps to build a bond with your cat.
When a cat has hairballs, you can use hairball remedy, which is a common method. Hairball remedy can soften the hair knots in the cat's stomach and intestines, making it easier for the cat to vomit or excrete the hairballs accumulated in its body. Regularly feeding your cat hairball remedy can also effectively prevent and treat hairballs.
You can also grow some cat grass at home, such as wheatgrass, barley grass, oat grass, and foxtail grass. They all contain high levels of fiber. When cats suffer from hairballs, feeding them these can effectively alleviate the condition and help them expel the hairballs.
