From the Super Mario Wiki
Tweesters are tornado-like enemies found in many Mario games that try to touch Mario, sending him into the air in an attempt to redirect his movements and hinder his progression. In most of the games they cannot harm Mario directly and can only attempt to force him into a more dangerous situation. They are usually found in desert levels.
Super Mario series
Super Mario Bros. 3
Tweesters debut in Super Mario Bros. 3, only appearing in the level Desert Land, where they can trap Mario in an air pocket and throw him backwards if he gets too close. The only way to get past one is to run until the Power Meter is fully charged, and then jump into the air before colliding with the Tweester. There is no way for Mario to defeat a Tweester in this game, and thus they will always remain in the same location.
Super Mario 64 DS
In Super Mario 64 DS, if Mario comes into contact with one of them, he is thrown upwards and slowly floats downwards (similar to jumping off a Spindrift). They are also situated next to quicksand, so the player must be careful not to fall in. In these games, they only appear in Shifting Sand Land.
Super Mario Sunshine
New Super Mario Bros.
In New Super Mario Bros., Tweesters appear in World 6-A. They are very large and immobile, and they no longer appear with threatening faces. If Mario comes in contact with one, he is thrown upwards and spins as he floats back down (similar to Super Mario 64).
Super Mario Galaxy
Tweesters appear in Super Mario Galaxy; however, instead of having menacing faces, they only have white eyeballs that follow Mario around. These Tweesters can also be helpful at times, being used to traverse obstacles. However, the garbage they are flinging around can hurt Mario or Luigi if they make contact with it. Touching one will also cause them to spin around. If the player then spins using the Wii Remote, the Tweester will make him go flying into the air, briefly making Mario or Luigi spin their arms around very quickly like a helicopter. Similar, but larger tornadoes without eyes known as Galactic Tornadoes also appear in this game. These tornadoes appear in the Dusty Dune Galaxy and also make a small appearance in Bowser's Galaxy Reactor. Tweesters do not appear in the sequel, Super Mario Galaxy 2.
The Adventures of Super Mario Bros. 3
In Paper Mario, the first game in which they have menacing faces, they forcefully blow Mario one screen to the north and one screen to the east in Dry Dry Desert if he comes into contact with them. It is highly recommended not to touch them, as the player can easily get lost within the vast sands. However, one particular Tweester (in an area with blue cacti) can be used to throw Mario next to an elevated Badge on top of a high rock which cannot be reached by any other means.
Mario Party series
Like several other minor enemies, Tweesters were featured as Capsules in the Mario Party franchise. In Mario Party 5, a Tweester Capsule can be used to send the Star to a new location. It has the same effect as the Lucky Lamp in Mario Party 3 and is also similar to the Chomp Call in Mario Party 4.
In Mario Party 6 and Mario Party 7, players can obtain a Tweester Orb that can be placed on the board as a roadblock. If other players pass through its space, a Tweester appears and moves the player to a random space on the board. It is also possible that the Bloway Candy in Mario Party 8 is inspired by Tweesters, as players who use it become a tornado and forcibly throw any other players they come into contact with back to the Start Space. This function is also somewhat similar to the Klepto Orb from Mario Party 6.
Dance Dance Revolution: Mario Mix
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