Yes, those are both possibilities for building a polarizer [1].

For example, a simple polarizer could be a grid of thin metal wires whose spacing is smaller than the wave-length of the incoming light. For the component of the E-field parallel to the wires currents can be induced freely along their length, and so the grid behaves much like a solid metal plate and reflects that part of the wave. For the component of the E-field perpendicular to the wires, significant currents can't be generated (since the wires are thin) and that part of the wave passes through.

[1] https://en.wikipedia.org/wiki/Polarizer

[2] https://en.wikipedia.org/wiki/Polarizer#Wire-grid_polarizers

> Where goes the energy of the orthogonal component of the field? Absorbed by the polarizer, reflected, ... ?

It depends on the type of polariser.

The type used in LCD displays and 3D cinema glasses absorbs, that's why everything looks darker through them but they don't look like mirrors.

A polarising beam splitter reflects one mode and passes the other. It looks like a half-mirror.