  Given a target distribution $\mu$ and a proposal chain with generator $Q$ on
a finite state space, in this paper we study two types of Metropolis-Hastings
(MH) generator $M_1(Q,\mu)$ and $M_2(Q,\mu)$ in a continuous-time setting.
While $M_1$ is the classical MH generator, we define a new generator $M_2$ that
captures the opposite movement of $M_1$ and provide a comprehensive suite of
comparison results ranging from hitting time and mixing time to asymptotic
variance, large deviations and capacity, which demonstrate that $M_2$ enjoys
superior mixing properties than $M_1$. To see that $M_1$ and $M_2$ are natural
transformations, we offer an interesting geometric interpretation of $M_1$,
$M_2$ and their convex combinations as $\ell^1$ minimizers between $Q$ and the
set of $\mu$-reversible generators, extending the results by Billera and
Diaconis (2001). We provide two examples as illustrations. In the first one we
give explicit spectral analysis of $M_1$ and $M_2$ for Metropolised independent
sampling, while in the second example we prove a Laplace transform order of the
fastest strong stationary time between birth-death $M_1$ and $M_2$.
