An engine having only one valve is known as a monosoupape engine. The idea was first used in World War 1 in aircraft with the specific intent of making rotary engines more reliable. (At first, it sounded about right; having only one valve is more reliable than having two valves.) Gnome was one of the first companies to manufacture these monosoupape engines, and they were used by the Allied Forces in WW1. In case you don't know, a rotary engine is often used today to refer to a wankel stroke engine, but it also refers to a radial engine that essentially has its mounting inversed; the crankshaft is bolted to the aircraft, and the propeller is bolted to the engine block. (For this post, whenever I say "rotary engine," I mean this type of radial engine, NOT the wankel stroke engine.) This mounting configuration has several advantages, including improved air cooling and a strong breeze of fresh air to each valve. It also eliminated the need of a flywheel. Because of the centrifugal force on rotary engines, valves failed frequently. When only one valve occupied the space of the cylinder head, it was easier to make that one valve beefier than each of the would-be two valves.
Because the rotary engine design had a strong breeze of fresh air to the cylinders, and because it didn't need exhaust pipes, it would sound like a perfect idea to use only one valve. As the exhaust exited the valve, it would get immediately blown away so fresh air could enter. Like a two-stroke engine, fuel entered through a port in the sidewall of the cylinder from the crank case, but like a four-stroke engine, fuel entered the chamber only every other stroke.
However, there were soon discovered disadvantages to this type of engine. First, the throttle control was primitive. The ignition was turned on for full power, and turned off for no power. Cutting fuel supply would result in a very slow throttle response, so cutting ignition was preferable. If the ignition was left off during flight, unburnt gasoline would accumulate on the sidewalls of the engine cowling. Once ignition was restored, any fuel in the cowling would catch fire from the outgoing hot exhaust. This meant pilots had to keep the power off for only short periods of time, otherwise the whole plane may be engulfed in fire. Needless to say, it was also a huge waste of fuel. The idea was eventually abandoned, and engines today now have two valves.
Long story short, it is a bad idea to mix exhaust fumes and intake fumes together. This is why we segregate them into two valves. If you like to know more about monosoupape engines, do a web search of it. Other sites explain it much better than I do.