We present two families of two-sided non-equivariant confidence intervals for the mean $\theta$ of a continuous, unimodal, symmetric random variable that, compared with the conventional, symmetric, equivariant confidence interval, are shorter when the observation is small, and restrict the sign of $\theta$ for smaller observations. One of the families, a modification of Pratt's (1961) construction of intervals with minimal expected length when $\theta=0$, is longer than the conventional symmetric interval when $|X|$ is large, and has longer expected length when $|\theta|$ is large. The other family gives the conventional symmetric interval when $|X|$ is large, with a change to the smaller endpoint when $|X|$ is small. Its expected length is less than that of the conventional symmetric interval when $|\theta|$ is small, larger for an intermediate range of $|\theta|$, then approaches that of the conventional interval for large $|\theta|$. This slight modification of the conventional two-sided interval has most of the power advantage of a one-sided interval, but short length.