Correct me if I am wrong, but aren't digital clocks run from a crystal for the timebase and use all DC voltages internally?
Virtually all line-powered clocks use the line frequency as a time base, not a crystal oscilator.
This is because even, say, an error of 50 PPM (i.e. a moderatly GOOD crystal oscilator) will gain or lose 4 1/3 seconds per day. Customers would complain and pitch it out.
Meanwhile, the power company adjusts the line frequency continuously so that a synchronous motor attached to a clock face by a gear train will hold against the national bureau of standards timebase within a small fraction of a second as long as it is powered continuously.
Indeed: Using power to distribute accurate time to cheap synchronous-motor clocks was one of the original selling points for AC, and planned before the sale of the first hookup (if I recall my history correctly).
If you want to power a clock off an inverter (which has rotten regulation as a timepiece even if it IS sine wave) and have it keep accurate time, get one of the "atomic clocks" - actually a crystal-oscilator clock with a radio that updates it from one of the WWV broadcasts.