  The (total) connected domination game on a graph $G$ is played by two
players, Dominator and Staller, according to the standard (total) domination
game with the additional requirement that at each stage of the game the
selected vertices induce a connected subgraph of $G$. If Dominator starts the
game and both players play optimally, then the number of vertices selected
during the game is the (total) connected game domination number ($\gamma_{\rm
tcg}(G)$) $\gamma_{\rm cg}(G)$ of $G$. We show that $\gamma_{\rm tcg}(G)\in
\{\gamma_{\rm cg}(G), \gamma_{\rm cg}(G) + 1, \gamma_{\rm cg}(G) + 2\}$, and
consequently define $G$ as Class $i$ if $\gamma_{\rm tcg}(G) = \gamma_{\rm cg}
+ i$ for $i \in \{0,1,2\}$. A large family of Class $0$ graphs is constructed
which contains all connected Cartesian product graphs and connected direct
product graphs with minumum degree at least $2$. We show that no tree is Class
$2$ and characterize Class $1$ trees. We provide an infinite family of Class
$2$ bipartite graphs.
