Learn Spanish (Latin America): Este logotipo está torcido torcido.

Repeating the word is a common spoken trick called lexical reduplication. It is used to express the absolute maximum degree of a quality. While muy torcido just means very crooked, saying torcido torcido emphasizes that it is unmistakably, completely crooked with no room for debate.

Does the second torcido have to agree in gender and number with the noun?

Yes. Because you are simply repeating an adjective, both instances must match the noun they modify. If you were talking about lines (las líneas), you would say están torcidas torcidas.

Why is it está torcido instead of es torcido?

We use está because torcido (crooked) describes the current physical condition of the logo, implying it was placed or printed incorrectly. If you said es torcido, it would mean the logo was deliberately designed to be crooked as a permanent, defining characteristic.

Is logotipo the most natural word to use, or can I just say logo?