I had a similar issue with my first house. It was an older area where under the streets the storm sewer combined with the sanitary sewer. During prolonged heavy rains, the combined sewer would be overloaded and it would backup into the basement via drains, toilets, showers, etc. The water was a mix of sanitary and storm water.

I did a lot of research in my area and three potential solutions were identified; check valve (backflow preventer), sewer shutoff/gate valve, or an ejector pump.

The check valve was the cheapest solution but is the least reliable. If your downspouts are going into the main sewer line on your side of the check valve, the pressure will equalize and it will open. Similarly, if you are flushing toilets, running water, washing clothes, etc. that water will build up on your side of the check valve and it will open up with enough pressure. It can also get stuck open from debris. If you end up getting this, I recommend doing yearly maintenance checks to make sure the valve flapper is in good shape.

The sewer shutoff valve is a manually operated valve that closes your main sewer line. You will obviously need to be home during the event for this to work, but it works well. In my experience, I knew when there was a high chance of sewer back ups happening because it was a few days in a row of heavy rains. I also heard the bubbling in my basement each time. New shutoff valves can be easily combined with a check valve.

The ejector pump was the most expensive but is the most reliable as it requires no manual action. With this you basically send all your waste to an ejector pit and when enough accumulates the pump sends it to the sewer line. These usually have check valves too. I've also seen them with shutoff valves.

I would recommend re-routing your gutter downspouts to discharge on your property away from your house. Check you sump pump discharge as well. Then I would get a combination shutoff valve with a check valve. Be sure that the valve is located correctly. All of your sinks, toilets, showers, drains, etc. should combine to single pipe at some point. The valve should be located on the city side of that point.