A local martingale X is called arithmetically symmetric if the conditional distribution of XT-Xt is symmetric given , for all 0<=t<=T. Letting , the main result of this note is that for a continuous local martingale X the following are equivalent: (1) X is arithmetically symmetric. (2) The conditional distribution of XT given is N(Xt,<X>T-<X>t) for all 0<=t<=T. (3) X is a local martingale for the enlarged filtration for each T>=0. The notion of a geometrically symmetric martingale is also defined and characterized as the Doléans-Dade exponential of an arithmetically symmetric...</=t<=t.></x></=t<=t.>