Pairing numbered gloves within one of each other

Ten gloves carry numbers, with two gloves labelled each of $1,2,3,4,5$. The gloves are split into $5$ pairs uniformly at random. What is the probability that in every pair the two numbers differ by no more than $1$?

$\tfrac1{45}$. There are $\tfrac{10!}{2^5\,5!}=945$ pairings. For the good ones, one glove labelled $n$ is paired either with its twin (leaving $n-1$ labels) or with one of the two gloves labelled $n-1$, which forces the other pair, so $g_n=g_{n-1}+2g_{n-2}$. From $g_0=g_1=1$ this gives $g_5=21$ and probability $\tfrac{21}{945}$.