Symmetry or Dynamic Consistency?

In a setting with repeated experiments, where evidence about the experiments is symmetric, a decision-maker ranks bets (or acts) over their outcomes. We describe a stark modeling trade-off between symmetry of preference (indifference to permutations), dynamic consistency and ambiguity. Then, assuming that experiments are ordered in time, we outline an axiomatic model of preference that exhibits dynamic consistency and yet models learning under ambiguity.