 Planar Brownian stochastic area

For independent coordinate Brownian motions, the stochastic area with the stated orientation is $Z_t={\int }_0^t{Y}_{s}d{X}_s-X_{s}d{Y}_s$. Its bracket is ${\int }_0^t(X_s^2+Y_s^2)ds$. Reversing orientation changes its sign but not its law or quadratic variation.