In this paper we study a seemingly unnoticed property of bases in a Hilbert space that falls in the general area of constructing new bases from old, yet is quite atypical of others in this regard. Namely, if $\{x_n\}$ is any normalized basis for a Hilbert space $ H $ and $ \{\,f_n\}$ the associated basis of coefficient functionals, then the sequence $\{x_n+f_n\} $ is again a basis for $ H $. The unusual aspect of this observation is that the basis $ \{x_n+f_n\} $ obtained in this way from $\{x_n\} $ and $ \{\,f_n\} $ need not be equivalent to either, in contrast to the standard techniques of constructing new bases from given ones by means of an isomorphism on $ H $. In this paper we study bases of this form and their relation to the component bases $ \{x_n\} $ and $ \{\,f_n\}$.