Even vs. Odd Heads: Does Symmetry Force Fairness?

You flip a coin 10 times. The coin has an unknown probability $p$ of landing heads on each flip, and the flips are independent. Suppose you are told that the probability of getting an even number of heads (0, 2, 4, 6, 8, or 10) is exactly equal to the probability of getting an odd number of heads. Does this guarantee the coin is fair -- i.e., must $p = 1/2$?