In “Trailing a dovetail shuffle to its liar” by Diaconis and Bayers. In section 2, he says the following.
Cutting a deck of cards respect the cyclic order of deck of cards, where card 1 follows card n. If we imagine a deck of cards in a loop, then cutting the deck rotates the loop.
What do these two statements mean ?
I don't really know what you're having issue with, so all I can do is expand those two sentences into more sentences saying the same thing. Hope it helps.
Suppose we have a deck of $n$ cards numbered $1,2,\ldots, n$, which is sorted in order. If we cut the deck, the new order will be $$ i, i+1, i+2,\ldots, n-1, n, 1, 2, \ldots, i-2, i-1 $$ They describe this as "respect the cyclic order" because if you imagine laying out all the cards in order along a circle both before and after the cut, then nothing has really changed. As you go around the circle, the cards lie in exactly the same order before and after the cut.
If you start laying out the circle at the top each time, then there is a difference: one circle is rotated with respect to the other. One has $1$ at the top, while the other has $i$ at the top.