Most of these games are older titles. Although there are a few more recent games that use a strange control method for the sake of challenging the player, these days they tend to be niche titles like Donkey Kong: King of Swing, which only uses shoulder buttons.
1. Dual Joystick (Shooting)
Representative games: Robotron: 2084, Inferno, Smash TV, Total Carnage (Williams, arcade), Space Dungeon (Taito America, arcade), Front Line (Taito, arcade), Geometry Wars and sequels (Bizarre Creations, Xbox 360, DS and Wii)
There are two joysticks on the control panel. The left stick moves the player around, and the right one determines the direction of fire. That means there is no fire button; the act of pushing the right stick signifies intent to shoot, if the game doesn't actually have automatic fire.
Moderate to high. Most controllers these days have two analog sticks. Best of all, mind, would be two digital sticks, especially for Robotron.
The scheme in use:
In creating the dual joystick control style, Eugene Jarvis solved a problem with 2D run-and-gun shooting games that has since become so ingrained into the genre that it seems strange to think it was fixed so early. Since Robotron: 2084, there's been Commando, Ikari Warriors, Gauntlet, Alien Syndrome, Zombies Ate My Neighbors, Gain Ground, Mercs, and Gauntlet Legends, along with scores more. All of these games support player facing only because of the need to specify what direction shots will go, so in all of them, in order to shoot in a direction the player must first move in that direction, unless (like Gauntlet), the player can't shoot while moving at all.
This might seem at first to be a control style for ambidextrous people, and there is a certain head pat/tummy rub aspect to it at first, but it's not hard to get the hang of. Most importantly, it makes the act of shooting at enemies while running from them much easier. All of these games feature hordes of monsters following the player constantly. In a game with the more common scheme, where the player must move the joystick to change facing, fire, then resume running, shooting behind always slows the player down.
It is going against the grain of current design trends to recognize that all player ability is ultimately granted. It is not always true that the player should be able to do everything he'd expect he could in real life. For more modern games this must usually be explained, but for older, more abstract games this is not as important.
This said, I know of no game on the list of button-shooters above that gains much from forcing the player to jerk the joystick backwards to fire at approaching opponents, and some (especially Commando) would be greatly improved with Robotron controls.
2. Joystick and Dial
Representative game: Aztarac (Centuri, arcade)
The control panel contains a dial and an analog joystick. Moving the joystick controls position. A trigger on the joystick controls firing, a little unintuitively, since it's the dial that aims the gun in any direction.
Poor. Attempts to map the dial to an analog stick could work, but would turn the control scheme into Robotron redux. It's a little more difficult to aim with a dial, but it could potentially allow for far more precise shooting than a joystick, digital or analog, could provide.
The scheme in use:
This one's a bit obscure. Aztarac is a very rare Centuri vector arcade game that Wikipedia tells us saw production in that fateful year, 1983. It's got extra-sharp graphics for a vector game, and is a tremendous challenge to play. KLOV states that only 500 machines were made, making playing Aztarac as its designer, the late Tim Stryker, intended nearly impossible for most people.
The player's spaceship can fly freely in eight directions with the joystick, and can aim in many more directions using a dial to control aiming. The object of the game is to roam around a huge area of space looking for enemy ships to destroy before they can fly in close enough to destroy a star base, destroying it. Contact is all that's required to destroy a base, and after the first couple of levels enemy ships begin flying in very rapidly and in ever-wider formations.
Although the player has infinite ships, losing all four bases means the end of the game, and before too long it becomes very challenging to get even one base through the attacks. After three or four levels, enemy formations become so large that they won't entirely fit on the screen at once, forcing the player to both fly around to find the ships and use the dial to shoot them when they show up.
Unlike Robotron, which is more-or-less constant shooting, Aztarac has long periods of silence while the player either flies out to meet the aliens or waits for them to arrive. When found, the player must take care of large groups of them very rapidly. The act of "hosing them down" is fairly well-suited by the dial aiming control. One thing Aztarac -- and for that matter Geometry Wars, which uses an analog stick to handle aiming -- have above Robotron is that the player can shoot freely in all directions.
Aztarac does have a couple of problems with its control. In addition to a stick and a dial there are two buttons: the fire button, which would be impossible to use effectively if it wasn't on the joystick, and a radar button. The game's pace is fast enough later on that taking a hand off the dial and pressing the radar button can be fairly disruptive, especially since it doesn't seem to pause the game while it's held.