Was flipping channels over the weekend and came upon a flick where someone was driving an automobile in reverse and seemed to be going at a high rate of speed.
This got me to thinking. Can a car go really fast in reverse? Would it shift gears in reverse, the same as if it were in drive? Seems to me that there would be a maximum MPH a car can to in reverse gear.
You could, if you don’t mind your engine leaping out of your hood in an ugly mess. Reverse in most cars has a gear ratio slightly lower than first gear. See how fast you can go in first before redlining. Reverse will be a bit slower than that.
Install the transmission backasswards. 
Most normal cars only have one reverse gear, so no, they will not shift.
Almost all of the driving in reverse that you see is very slow, probably <5 mph. Watch a cop or cabbie back up and they sometimes seem to be moving quite fast, and I think that’s due, for the most part, to most observers not being used to seeing cars back up at 20 mph.
It will also be a little tricky to steer. Driving forward, the steering has a tendency to return to the center. When driving backwards, that centering tendency is also reversed; turn the steering wheel a little bit and the forces will want to make it turn even more. You can overcome that with a firm grip on the wheel, but I imagine it would take a bit of practice to do it at higher-than-normal speed.
As Robot Arm said, there is a castor [US: caster - I think] principle in which the steering connection is set back slightly from the hub of the front wheels. This is why when you let go of the steering wheel travelling forwards, the car will tend to find a straight line. In reverse, the castor principle is the opposite, making the car behave like a shopping cart. Let go of the wheel in mid-curve in reverse, and rather than finding centre, the steering will tend to find full lock in the direction you originally had the curve.
I freak people out sometimes because I tend to reverse fast, and use the reverse castor principle to slot into spaces by simply letting go of the wheel at just the right time. It takes a bit of practice, and the reason I can do it is because I drive forklifts. Forklift trucks have this type of “rear end swing, reverse castor” steering when going forwards, as they are designed for high maneuvrability, not speed (the two don’t go together). A decent forkie can easily get a forklift in and out of a space literally no bigger than a couple inches longer than the length of the unit. This can’t be done in cars because that is the trade-off the manufacturers made so you can drive the thing safely at freeway speeds without flying all over the place. It’s also why you can’t drive your car forwards into a tight spot.
BTW, there was an Australian car manufacturer in the 1960s - I forget the name - (which, BTW, didn’t survive because their cars were crap, and they were better suited to their main business of making washing machines). These cars had a reverse “switch”, not a gear. To reverse, you’d turn off the ignition, flick the switch, turn on the ignition again, and the engine would rotate in the opposite direction. This of course, meant you had four reverse gears, and could theoretically achieve highway speeds going backwards.