Yes, this technique is known as redscale. You can find a decent amount of information by googling using that term, and see example photos on Flickr.
What you need to do
Technically this would be easiest to do on large format film, as all you'll need to do is to load the sheet the wrong way around. But experimenting with large format film in this way would also be incredibly expensive.
On 35mm you'll need to figure out a way of loading the film cassette so that the film inside is inverted – you cannot simply invert the cassette in the camera, since most (all?) cameras will not physically allow this. (If you know of a camera that does, please let us know in the comments!) It used to be possible to buy C41 film in bulk (e.g. a 30 m roll), which you would then cut and load onto a reusable cassette yourself. Bulk C41 film seems to be in short supply these days, so what you can do is to sacrifice an ordinary cassette by pulling the film out and cutting it, and then load it backwards onto the reusable cassette. Alternatively, you can tape the inverted film onto the stub in the old cassette and spool it back. Note that all of this has to be done in complete darkness (darkroom, changing bag), otherwise the film will be fogged and rendered useless.
Alternatively, Lomography makes a redscale film that is already inverted, so you load it just like any other film. However, it is out of stock as of this writing.
Why this works
Here's a very simplified cross-section of typical C41 colour negative film:
Light enters (normally) from the top. Basically the film consists of three layers of silver halides and dyes, each layer sensitive to different parts of the visible spectrum. (In processing, the silver eventually gets washed away and what remains is three layers of dye in yellow, magenta and cyan, forming the final colour negative.) It turns out that all layers are somewhat sensitive to blue light, which is why a yellow filter layer is sandwiched between the blue-sensitive and the other layers. This keeps most of the blue light from entering the green- and red-sensitive layers.
Now, if you invert the film, the order in which the layers are traversed is also inverted. Most notably, blue light now cannot reach the blue layer because of the yellow filter, meaning that blues tend not to be registered in the final image.
Light also needs to pass through the film base and the anti-halation layer before reaching the silver halide layers. (The anti-halation layer is there to keep light from reflecting back onto the light-sensitive layers from the boundary of the film base.) In practice, this means that exposure may need to be adjusted relative to the film's nominal speed, i.e. you may need to expose more than usual. It also means that it is possible to "redscale" black and white film, because even though it is just one light-sensitive layer, the anti-halation backing and the film base are still there. Of course, this does not mean your B&W images turn red, just that the image will be affected in some way depending on what the exact properties of the anti-halation backing and the film base are.
Is it worth it?
There is the argument that any analog effect, including redscale, can be reproduced digitally to any desired degree of perfection. This is probably true. However, the analog thing can be a fun experiment to conduct. Personally, I wouldn't be tempted to do this, at least not on a regular basis, unless I also did my own C41 processing, since lab costs pile up on whatever money is lost already on film stock while experimenting. (I also suspect labs might charge a premium for film that arrives in anything but an off-the-shelf, undoctored cassette, though am not sure.)