Is there a word that means the 'inverse of variance'? That is, if $X$ has high variance, then $X$ has low $\dots$? Not interested in a near antonym (like 'agreement' or 'similarity') but specifically meaning $1/\sigma^2$?
$1/\sigma^2$ is called precision. You can find it often mentioned in Bayesian software manuals for BUGS and JAGS, where it is used as a parameter for normal distribution instead of variance. It became popular because gamma can be used as a conjugate prior for precision in normal distribution as noticed by Kruschke (2014) and @Scortchi.
Kruschke, J. (2014). Doing Bayesian Data Analysis: A Tutorial with R, JAGS, and Stan. Academic Press, p. 454.