I will discuss two related questions concerning the two spaces in the title: the Čech-Stone remainder of the natural numbers $\mathbb N$, and the Čech-Stone remainder of the half-line $\mathbb H = [0,\infty)$. Both $\mathbb N$ and $\mathbb H$ are naturally ordered from left to right. These orders on $\mathbb N$ and $\mathbb H$ are reflected in their Čech-Stone remainders, in certain dynamical systems on $\mathbb N^\ast$ and in certain subcontinua of $\mathbb H^\ast$. Are these left-to-right aspects of $\mathbb N^\ast$ and $\mathbb H^\ast$ truly topological, or can either of the spaces be “reversed” via some self-homeomorphism?