Two directional FET switch. Is it possible?
Fun fact: MOSFETs (JFETs too!) will conduct in either direction when Vgs is above threshold.
The issue with MOSFETs (3-terminal ones, with the body tied to source) is that the parasitic body diode that's formed by the FET structure and body-source tie also conducts when the drain-source voltage is in the reverse direction even if the FET is off.
More about that pesky body diode here: How should I understand the intrinsic body diode inside a MOSFET?
In other words:

FET 'on': conducts in both directions through drain-source channel with Rds(on) resistance
FET 'off': blocks forward drain-source, conducts reverse through body diode with one Vf drop.

Note that when 'on', the FET drain-source channel shorts out the body diode, so the large majority of the current flows through the normal drain-source path and not the parasitic diode.
The workaround is to tie two FETs in series, back-to-back, source-to-source or drain-to-drain. This allows each FET to block the other's parasitic body diode when 'off'.
More here: Does MOSFET let current flow through source to drain as it allows it from drain to source?
BONUS: A simulation showing high-side switches built with n-FETs in 3 different ways: 4-terminal, 3-terminal with body diode, and back-to-back 3-terminal.
Simulate it here