This is not a complete answer, but it was to long to fit into a comment.

I'm having troubles with the same question. I haven't found a good explanation about this anywhere yet.

So far what I managed to realize is that if we "fold" the band for this system $n$ times we get a set of eigenvalues $2 t \cos(ka + (2 m \pi / n))$, $m = 0, ..., n-1$, which is a generalization of the result you had above. This can be interpreted as n equivalent shifted bands, each centered around $2 m \pi / n$.

This leads me to think of the "folding/unfolding" of the bands as actually just shifts in momentum space, but I'm not sure if this interpretation is correct or if it breaks down at some point.

I got dragged into this topic by reading this article, where they some some visualization techniques for band unfolding. You could find it or its references useful somehow.

Hopefully I'll update this answer when I find a more satisfying discussion on the topic.

This is not a complete answer, but it was too long to fit into a comment.

I'm having troubles with the same question. I haven't found a good explanation about this anywhere yet.

So far what I managed to realize is that if we "fold" the band for this system $n$ times we get a set of eigenvalues $2 t \cos(ka + (2 m \pi / n))$, $m = 0, ..., n-1$, which is a generalization of the result you had above. This can be interpreted as n equivalent shifted bands, each centered around $2 m \pi / n$.

This leads me to think of the "folding/unfolding" of the bands as actually just shifts in momentum space, but I'm not sure if this interpretation is correct or if it breaks down at some point.

I got dragged into this topic by reading this article, where they some some visualization techniques for band unfolding. You could find it or its references useful somehow.

Hopefully I'll update this answer when I find a more satisfying discussion on the topic.