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Inverted sprites (also known as Scrambled sprites or Reversed sprites) is a side-effect of encountering certain Glitch Pokémon or viewing their stat screen in the Generation I games.
When one first enters the battle, the trainer sprite is "glitchy" in appearance. It is also partially reversed from its normal state:
After the battle starts, one's Pokémon also have the "glitchy" back sprites. Note also that the enemy Rattata
also has its sprite reversed:
Once one uses an attack, the enemy Pokémon's sprite breaks as well.
As the name implies, this glitch causes the sprites to appear backwards or flipped. This x-mirroring effect is used in the stats screen normally, but is only intended for front sprites and does not react well with other things. For instance, the trainer sprite and back sprites get a "glitchy" appearance when flipping is attempted. In addition, while the front sprites flip properly originally, once the sprite is updated (through attacking), it breaks. The effect causes no permanent damage on the game however.
To fix the effect, the Pokédex entry or stats screen of a non-glitch Pokémon needs to be viewed.
Official Nintendo statement
Nintendo has an official description of MissingNo. listed in their Customer Service troubleshooting section, which includes a nonworking method of fixing the effect:
"MissingNO is a programming quirk, and not a real part of the game. When you get this, your game can perform strangely, and the graphics will often become scrambled. The MissingNO Pokémon is most often found after you perform the Fight Safari Zone Pokémon trick.
To fix the scrambled graphics, try releasing the MissingNo Pokémon. If the problem persists, the only solution is to re-start your game. This means erasing your current game and starting a brand new one." 
Nintendo was incorrect on the method for fixing this behavior; releasing MissingNo. does not fix inverted sprites. Instead, looking at the stats of a non-glitch Pokémon should fix the problem.
In Red, Blue, and Yellow, an address in the RAM determines whether the sprite of the Pokémon in battle is inverted or not. The address in Red/Blue is 0xd0aa, and 0xd0a9 in Yellow. The normal value is 0x00, which results in the sprite presented in its normal orientation. When any value other than 0x00 is entered, it results in the sprite being inverted.
Viewing the Pokédex entry of glitch pokémon with the Pokédex number 00 or a number above 151 results in the value being changed, therefore inverting the sprite. Viewing the Pokédex entry or stats screen of a non-glitch Pokémon resets the value to 0x00 and makes sprites be presented in the normal orientation.