Detecting the Direction of Time in a Price Series

You are handed a time series of a stock's daily closing prices, but someone may have reversed the time axis -- flipping the series end-to-end so the last day appears first. Your job is to figure out whether the series is running forward or backward. Here is the catch: if the price followed a pure Brownian motion (i.e., increments are i.i.d. Gaussian), the distribution of the path looks identical forward and backward. So you cannot rely on anything that holds for a symmetric random walk. What statistical properties of real financial data break time-reversibility, and how would you use them to detect whether the series has been flipped?