According to the Alphabetilately website, "A bisect is a postage stamp cut in half (usually diagonally), and used to pay half its face value, e.g. half a ten cent stamp to pay five. The practice has been permitted (in the US at least) only for special situations (e.g. a shortage of stamps). The most recent of them was over 60 years ago, so genuine bisects are usually rare and valuable. There are even trisects! Note that a bisect "off cover" (or not "tied" to the cover by its cancellation) is usually worthless, since it is the proof of actual usage that makes it desirable - anyone can cut a stamp in half, but if it survives the mails that way, it becomes something special. Don't try it today - it's illegal to cut, deface or even overlap stamps on your mail."
Shown above is an extreme example of bisected stamps which has an interesting story behind it.
Alphabetilately webmaster, William M. Senkus, wrties "There's no pretense of postal validity about this cover, a souvenir prepared by the author at Pacific 97. I call it my "ultimate bisect cover". One of the odder pasttimes of collectors at philatelic events is the creation of such souvenirs, though most are more conventional First Day and event covers. I had to stand in line for almost an hour just to reach the counter with this one, and then was told that the stamps, having been cut in half, were no longer postally valid, and even though the cover would never pass through the mails, it could not be cancelled! I pointed out that there was over $2.50 in postage on the envelope, and surely it would do no harm to cancel it. Twenty minutes and several levels of consultation later, I was told it could be cancelled, though not in the manner I had requested - I forget now exactly what the changes were, but was pleased with the result, as much for the commotion it caused as for the actual product."
Senkus goes on to say, "I have just (October, 2001) been informed by a fellow collector who saw my page of Pacific 97 souvenirs that the USPS actually authorized bisects of stamps from the two USPS souvenir sheets issued at the show. He says that the slightly odd conditions were that the cuts must not separate the digits of the denominations, and the bisects were valid only during the show. So diagonal separations were ok, and horizontal separations were ok, but vertical separations (cuts from top to bottom down the middle) were not. Since I was not told about that when I submitted the cover above for cancellation, I can only presume that if this is true, the policy was revealed to the public only after the show; so if one thought to try it, one succeeded, but one never knew whether that success was a fluke or deliberate.
"And this could help explain, perhaps, why my cover caused such a stir among the clerks and required a secret backstage consultation - If they had said the bisects were totally illegal, they would have been lying, while if they had told me the problem with my cover was that I had bisected the triangle stamps, they would have been revealing that the other bisects were ok. So perhaps they decided to let the cover pass as it was, and let me think the whole thing was a special favor."
For more on bisected stamps and covers, click here