  For each ordinal $\alpha<\omega_1$, we introduce the class of
$\alpha$-balanced Polish groups. These classes form a hierarchy that completely
stratifies the space between the class of Polish groups admitting a
two-side-invariant metric (TSI) and the class of Polish groups admitting a
complete left-invariant metric (CLI). We establish various closure properties,
provide connections to model theory, and we develop a boundedness principle for
CLI groups by showing that $\alpha$-balancedness is an initial segment of a
regular coanalytic rank.
  In the spirit of Hjorth's turbulence theory we also introduce "generic
$\alpha$-unbalancedness": a new dynamical condition for Polish $G$-spaces which
serves as an obstruction to classification by actions of $\alpha$-balanced
Polish groups. We use this to provide, for each $\alpha<\omega_1$, an action of
an $\alpha$-balanced Polish group whose orbit equivalence relation is strongly
generically ergodic against actions of any $\beta$-balanced Polish group with
$\beta<\alpha$.
