What is your basis of assuming there are (for any definition of 'right-wing') 'no right-wing vegans'? It seems like a baseless assertion. Adolf Hitler infamously identified as a vegetarian and tried to dissuade others from eating meat due to animal suffering (according to witnesses).[wiki]

• 'Right-wing' is extremely vague and doesn't imply an attitude towards animal wellbeing or environmental concerns (see: eco-fascism)
• 'Right-wing' doesn't even imply any (conscious!) attitude towards human suffering. It's very common for them to legitimately think they're doing the right thing and reducing suffering in the world, despite obvious counter-evidence (if they sincerely believe the mainstream US narrative that socialism will cause mass starvation and mass murderous government repression and genocide, then opposing it would seem morally better than the status quo). But, even if it did imply they were fine with human suffering, it could be seen by them as deserved (like a vengeance thing, or like anti-parasitism like squashing mosquitoes) as opposed to animal suffering for commercial production.

Despite all this: people hold inconsistent views and values all the time. This isn't rare or anything. It's actually extremely normal. I wouldn't even call it cognitive dissonance, it's standard neurological behaviour.

But at least Lifting Vegan Logic has said that he is neither left, or right. Usually I would say it means he is right-wing, so I am wrong?

The ideas of 'left' and 'right' are basically meaningless. They are inherently relative and vaguely defined to the point they can mean anything. Chances are, if they're being sincere, they don't consider themselves political and therefore believe that makes them neutral. Most of us would obviously disagree.