  Let $W$ be a Coxeter group, and for $u,v\in W$, let $R_{u,v}(q)$ be the
Kazhdan-Lusztig $R$-polynomial indexed by $u$ and $v$. In this paper, we
present a combinatorial proof of the inversion formula on $R$-polynomials due
to Kazhdan and Lusztig. This problem was raised by Brenti. Based on Dyer's
combinatorial interpretation of the $R$-polynomials in terms of increasing
Bruhat paths, we reformulate the inversion formula in terms of $V$-paths. By a
$V$-path from $u$ to $v$ with bottom $w$ we mean a pair $(\Delta_1,\Delta_2)$
of Bruhat paths such that $\Delta_1$ is a decreasing path from $u$ to $w$ and
$\Delta_2$ is an increasing path from $w$ to $v$. We find a reflection
principle on $V$-paths, which leads to a combinatorial proof of the inversion
formula. Moreover, we give two applications of the reflection principle. First,
we restrict this involution to $V$-paths from $u$ to $v$ with maximal length.
This provides a direct interpretation for the equi-distribution property that
any nontrivial interval $[u,v]$ has as many elements of even length as elements
of odd length. This property was obtained by Verma in his derivation of the
M\"obius function of the Bruhat order. Second, using the reflection principle
for the symmetric group, we obtain a refinement of the inversion formula by
restricting the summation to permutations ending with a given element.
