Idiots aside, Islamism is obviously far-right. Okay, left- and right- don't have good definitions as is, as they were defined in response to where people physically sat in the Estates General during the French Revolution. But I can at least point to a few properties that are manifestly right-wing and why Islamists all tick those boxes:
1. Right-wing politics supports and defends already-established power structures. Islam (Sunni-Shia divide notwithstanding) is an established power structure; in fact, it's the dominant power structure in the region. Ergo, Islamism supports and defends an established power structure: a specific religion.
2. Right-wing politics is Wilhoitist: it believes in an in-group, to whom the law protects but does not bind; and that the law binds but does not protect the out-group. Islamism specifically wants an Islamic theocracy, that's about as Wilhoitist as it gets.
3. Right-wing politics demonizes the other, to the point of spreading lies about people outside their preferred in-group. Whenever there's someone shitting on Islam you can count on Islamists to make shit up about them to stoke the flames of outrage.
To me, at least, those properties alone are enough to confidently put Islamism in the far-right bin. The fact that some ignorant leftists do the whole "the enemy of my enemy is my friend" bit without actually reading what's on the tin is immaterial.