In the land of the tetrominos, the twins L. and J. have set out to revolutionize travel by rail. Contemporaries' passenger cars are boxy and boring, and humdrum symmetric passenger arrangements are all too common when the car gets full. "Not on our lines!" vow L. & J. No, when travelers completely fill the polymino interior of an L. & J. railcar, the entrepreneurs promise that both reflection-symmetric and rotation-symmetric seating will be impossible by design.
That is, if they can only get one of their interiors approved by the railcar authorities, I., T., and O. Prior submissions have fallen afoul of one or more of the safety regulations:
Polymino cells must be axis-aligned. (By convention, the $\pm x$ directions are the car's directions of travel.)
The car should be balanced on the tracks; the interior must be symmetric about the $x$ axis.
The car should travel equally well in both directions; the interior must be symmetric about the $y$ axis.
Besides all that, L. and J., being good businessminos, don't like wasting space; it should be possible to fill their cars to the brim.
How can L. and J. build the railcar of their dreams? Bonus: What is the least capacity such a car could have?
(Since this is an aha-moment sort of puzzle, please spoiler your answers.)