(This is an excerpt from "Table Tennis Tactics for Thinkers.")
This grip was named for and popularized by five-time U.S. Men’s Singles Champion Dan Seemiller, who was ranked in the top thirty in the world in the late 1970s. He was followed by Eric Boggan, who reached top twenty in the world. No other U.S.-trained player has come close to these rankings in the sponge era (since the 1950s). Four of the five U.S. team members at the 1983 World Championships used this grip - Dan Seemiller, his brother Rick Seemiller, Eric Boggan, and 1983 Pan Am Men’s Singles Gold Medalist Brian Masters. (All four are in the U.S. Table Tennis Hall of Fame.) The grip is sometimes called the American grip, but is more commonly called the Seemiller grip. (Here's a video of Dan Seemiller, the lefty, in the final of the 1989 LA. Open.)
The grip is sort of a variation of the shakehands grip, with the top of the racket rotated to the left so that the index finger curls around the side of the racket. The forehand is played about the same, but on the backhand the arm rotates about so that the same side is used on the forehand and backhand. Despite its promising start, the grip never came close to the popularity of shakehands or penhold, and in recent years fewer and fewer players use the grip. However, you will face these players in tournaments (especially in the northeast U.S.) and need to be ready.
Like shakehands and penhold, the Seemiller grip also has its backhand and forehand variations, except here it is more extreme. If the top of the racket is rotated to the left, it is a backhand grip, as used by Eric Boggan and Brian Masters, which weakens the forehand loop. If the racket is rotated to the right (almost becoming a regular shakehands grip), it is a forehand grip, as used by Dan and Rick Seemiller, which weakens the backhand.
The Seemiller grip has four major advantages. It is probably the best grip for blocking, especially on the backhand. There is very little weakness in the middle - in fact, the grip is at its best there. It gives a very natural wrist snap on forehand loops against backspin. And since only one side of the racket is used, and because the racket is easy to flip with this grip, it allows a player to have an off-surface on selected shots, usually antispin, though some use long pips. A player with this grip can flip to use that side as a variation, and then flip back to the regular surface, usually inverted. (All four of the U.S. team members mentioned above used antispin on the reverse side, inverted on the other.)
The disadvantages are that the wrist can make it difficult to play the corners (and so players with this grip often have trouble with players who play to the wide corners); it limits the backhand mostly to close-to-the-table blocking and hitting, with a very limited backhand loop; and it can be difficult to counterloop with this grip.
Because of the lack of a strong backhand loop, deep serves and pushes to the backhand can give this grip problems, unless the player has very fast footwork and can play the forehand from the backhand over and over. (Others, like Eric Boggan, learned to hit backspin serves with his antispin side, and then flipped back to inverted for the next shot.)
Some players with the Seemiller grip can be absolute walls on the backhand, and it makes no sense trying to overpower that side - but if you attack the forehand side first (and perhaps force them a step off the table) and then come back to the backhand, then the backhand wall might crumble. The grip is weaker from off the table, and like most shots, is less consistent when you have to move.
Most players with the Seemiller grip use the off surface to return serves, especially short ones. Some have the ability to quickly judge the depth of the incoming serve, and use anti against short serves, inverted to loop or otherwise attack long serves. If they use the anti to return most serves, serve deep, and you should get a relatively weak return or an erratic anti attack. Often a deep serve to the forehand is especially effective. If they try to flip the racket based on the depth of your serve, mix in short spinny serves and fast, long serves, and watch them struggle to flip their racket appropriately - it’s not easy! It is very important not to telegraph your serves - players like this are very good at picking up small cues, so try to use the exact same motion for both short and long serves, at least until contact.