All right, I've been re-watching Legend of Korra from the beginning, because why would you not, and also Korrasami, and I'm trying to map out the different bending styles and abilities. Please let me know if I'm missing anything:

Earthbending

So, we've got regular-style Earthbending, as taught to humans by badgermoles.

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And then Metalbending, which Toph invented when she was trying to escape the bounty hunters.

And of course, Lavabending, which was first introduced in Book 3 of Korra through the character of Ghazan - but it probably existed before we actually met him. In the original Avatar series, an un-named Firebender Avatar is shown causing volcanoes to erupt.

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And Sandbending, which gave Toph some trouble when Team Avatar was visiting Wan Shi Tong's library of knowledge.

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Waterbending

There's regular Waterbending, as taught to humans by the moon, basically.

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Swampbending? Plantbending? Whatever these guys do.

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And of course, that nasty Bloodbending thing, which originally could only be performed during the full moon but that Yakone and his sons mastered and could employ regardless of the phases of the moon.

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Firebending

Regular Firebending, as taught to humans by the dragons. Some Firebenders can have hotter flame than others, which explains the color of Azula's fire.

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Lightningbending, which can be either produced or redirected by skilled Firebenders.

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UPDATE: I forgot the Combustionbending that P'Li and Combustion Man can do with their third eye.

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Airbending

Taught to the nomads by the bison.

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And flight, which existed only for the most enlightened and detached Airbending gurus before Zaheer performed the feat again, hundreds of years later.

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Am I missing any? I feel like I'm overlooking a form of Firebending, but these are all I can think of right now.