A symmetric smile says the market expects fatter tails on both sides than a normal distribution allows. It is the typical shape in currencies and many commodities, where a violent move up is roughly as plausible as one down.
Equities rarely smile; they smirk, with the put wing far above the call wing. When an equity chain does start to smile — both wings bid — it usually means the market is pricing a binary event where either outcome produces a jump, such as a takeover or a trial result.
Example: XYZ is a biotech at $50 awaiting a decision. The $40 put implies 78% and the $62.50 call implies 74%, while the $50 straddle implies 61%. Both wings are bid because the market expects XYZ to be somewhere other than $50 whatever happens.
Related: volatility-skew, volatility-surface, lognormal-assumption, earnings-play