We investigate positive integer sequences called throwback sequences, generated by moving the initial term of a given sequence to the right a number of places equal to its value, then repeating this step iteratively. Let X be a sequence of distinct positive integers. We prove that each term x of X appears infinitely often in the throwback sequence T(X) of X. Further, we provide an explicit formula for the limiting frequency with which x appears in X. If X is an increasing sequence, we prove that T(X) is uniformly recurrent, i.e., every block of consecutive terms in T(X) appears infinitely often with bounded gaps between consecutive appearances. We discuss how throwback sequences relate to familiar notions such as 2-adic valuations of natural numbers and the Gray code ubiquitous in modern telecommunications. Finally, we examine sorting and mixing properties of the iterated throwback operation in certain special cases.