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 $\mathbb{H}=[0,\mathrm{\infty })$. 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?