  Novel chirally symmetric fermion actions containing the minimum amount of
fermion doubling have been recently proposed in the literature. We study the
symmetries and renormalization of these actions and find that in each case,
discrete symmetries, such as parity and time-reversal, are explicitly broken.
Consequently, when the gauge interactions are included, these theories
radiatively generate relevant and marginal operators. Thus the restoration of
these symmetries and the approach to the continuum limit require the
fine-tuning of several parameters. With some assumptions, we show that this
behavior is unavoidable for actions displaying minimal fermion doubling.
