BOOOOOO !!!! This was just confirmed to be FALSE
"Just went to another legion and they are wrong dude. I have to wait 24 hours to invade. I do not see all the planets that they had posted. I do not see the planets, or NPCs, that were alerted before my arrival and finally the alerts from the legion I was just in is not available either because I had left their legion."
So yeah, apparently you do have to wait 24 hours to invade, you cannot join and invade immediately.
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EDIT / ADDITION: "Correction, I can invade enemy planets if alerted after I arrive it seems."
so... YAY ? It was partially right. You can invade immediately but only if the planet was alerted after you arrive and not before. FYI folks. If anyone has experienced it differently, feel free to post it.