  An edge-coloured path is \emph{rainbow} if all the edges have distinct
colours. For a connected graph $G$, the \emph{rainbow connection number}
$rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that,
any two vertices are connected by a rainbow path. Similarly, the \emph{strong
rainbow connection number} $src(G)$ is the minimum number of colours in an
edge-colouring of $G$ such that, any two vertices are connected by a rainbow
geodesic (i.e., a path of shortest length). These two concepts of connectivity
in graphs were introduced by Chartrand et al.~in 2008. Subsequently,
vertex-coloured versions of both parameters, $rvc(G)$ and $srvc(G)$, and a
total-coloured version of the rainbow connection number, $trc(G)$, were
introduced. In this paper we introduce the strong total rainbow connection
number $strc(G)$, which is the version of the strong rainbow connection number
using total-colourings. Among our results, we will determine the strong total
rainbow connection numbers of some special graphs. We will also compare the six
parameters, by considering how close and how far apart they can be from one
another. In particular, we will characterise all pairs of positive integers $a$
and $b$ such that, there exists a graph $G$ with $trc(G)=a$ and $strc(G)=b$,
and similarly for the functions $rvc$ and $srvc$.
