  We revisit and generalize the notion of dilation distance ${\rm d_{D}}(u,v)$
between unitary tuples and study its relation to the natural Haagerup-R{\o}rdam
distance ${\rm d_{HR}}(u,v) = \inf\{\|\pi(u) - \rho(v)\|\}$, where the infimum
is taken over all pairs of faithful representations $\pi \colon C^*(u) \to
B(\mathcal{H})$, $\rho \colon C^*(v) \to B(\mathcal{H})$. We show that ${\rm
d_{HR}}(u,v)\leq 10\operatorname{d_{rD}}(u,v)^{1/2}$, where ${\rm d_{rD}}(u,v)$
is a relaxed dilation distance, improving and extending earlier results. For an
antisymmetric matrix $\Theta$, we show via a concrete dilation construction
that a tuple of unitaries $u$ that almost commutes according to $\Theta$ (i.e.,
$\|u_\ell u_k - e^{i \theta_{k,\ell}} u_k u_\ell\|$ is small) can be nearly
dilated to a tuple of unitaries $v$ that commutes according to $\Theta$ (i.e.,
$v_\ell v_k - e^{i \theta_{k,\ell}} v_k v_\ell = 0$). We show that the dilation
can be "reversed" by a second application of the dilation construction, which
leads to a rotated version of the original tuple. Thus, a gauge invariant
almost $\Theta$-commuting unitary tuple can be approximated (in some faithful
representation) by a $\Theta$-commuting unitary tuple. Moreover, when $\Theta$
is ergodic, a $\Theta$-commuting tuple is shown to be {\em almost} gauge
invariant, and it follows from the results above that these can be approximated
in norm by $\Theta$-commuting tuples. In particular, we obtain the following
counterpart of Lin's theorem on almost commuting unitaries: if $q \in
\mathbb{T}$ is {\em not} a root of unity, then for every $\varepsilon >0$ there
exists $\delta > 0$ such that for every pair of unitaries $u_1,u_2 \in
B(\mathcal{H})$ for which $\|u_1 u_2 - qu_2 u_1\| < \delta$, there exists two
$q$-commuting unitaries $v_1, v_2 \in B(\mathcal{H} \otimes \ell^2)$ such that
$\|v_i - u_i \otimes 1\| < \varepsilon$ ($i=1,2$).
