In the Standard Model (SM) of particle physics, fermions are taken to be left handed. The reason is to incorporate the parity violation in weak interactions. Which is a fact of nature. If one check then in will be seen that $V-A$ type current can account for the inclusion of Left handed fermions ($V+A$ type current may also possible, but its ruled but by experiment).
Minimal extension of SM is the Left-Right symmetric model, proposed by Rabindra Mohapatra and Goran Senjanovic back late seventies, based on gauge group $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$. In this model the fermion fields are assigned to the doublets
$$L_{L}^{i}= \begin{pmatrix}\nu \\ e^- \end{pmatrix}_{L}^{i}\qquad L_{R}^{i}= \begin{pmatrix}\nu \\ e^- \end{pmatrix}_{R}^{i} \qquad Q_{L}^{i}= \begin{pmatrix} u \\ d \end{pmatrix}_{L}^{i} \qquad Q_{R}^{i}= \begin{pmatrix} u \\ d \end{pmatrix}_{R}^{i}$$
It is obvious from the field representations that both left and right handed fermions entered in the game with both hands. The transformation between $L$ and $R$ fields accomplished parity, and impose parity invariance before spontaneous breaking down $U(1)_{e}$.
This model has interesting features and anomaly free, and looks more symmetric compared to SM itself. But despite intense search, no data has found yet supporting this model.
An interesting paper published by Roni Harnik et al recently. Where authors considered the possibility of an universe without weak interactions. They provided theoretical arguments that it is indeed possible to have a stable universe with nucleosynthesis, matter domination, structure formation, in the absence of weak interactions, which is responsible for parity violation.