  We have recently proposed a two-dimensional quantum walk where the
requirement of a higher dimensionality of the coin space is substituted with
the alternance of the directions in which the walker can move [C. Di Franco, M.
Mc Gettrick, and Th. Busch, Phys. Rev. Lett. {\bf 106}, 080502 (2011)]. For a
particular initial state of the coin, this walk is able to perfectly reproduce
the spatial probability distribution of the non-localized case of the Grover
walk. Here, we present a more detailed proof of this equivalence. We also
extend the analysis to other initial states, in order to provide a more
complete picture of our walk. We show that this scheme outperforms the Grover
walk in the generation of $x$-$y$ spatial entanglement for any initial
condition, with the maximum entanglement obtained in the case of the particular
aforementioned state. Finally, the equivalence is generalized to wider classes
of quantum walks and a limit theorem for the alternate walk in this context is
presented.
