Can a beam splitter distinguish between more than two polarizations? The following is based on a very basic understanding of lasers, that
  may be approximate or altogether completely erroneous.
Given a (single-mode) laser beam, a beam-splitting polarizer distinguishes between two orthogonal polarizations that correspond to the optic axis of the birefringent material.  If the beam is s-polarized, it passes through, and if it is p-polarized, it is sent at an angle.  In the other cases, that is, with different polarizations, it is absorbed (is that correct?).
Question:  Does there exist a material/device that takes a laser beam as input, and sends it at an angle based on its polarization?  I know beforehand that the polarization is, e.g., a multiple of $\pi/6$.
Question (graphically):

 A: 
If the beam is s-polarized, it passes through, and if it is p-polarized, it is sent at an angle. 

This is correct

In the other cases, that is, with different polarizations, it is absorbed (is that correct?).

This is not correct. No matter the polarization state of the input (linear, elliptical, tilted..) the beam splitter will split the two linear orthogonal components of the input, which are parallel and perpendicular to the beam splitter axis. These two components are the projections of the input polarization state on orthogonal directions (a base) dictated by the orientation of the beam splitter. The sum of the two output intensities is always equal to the input intensity. For example, if the input is linear at 45 deg after the beam splitter you'll have two beams with identical intensity, one polarized linear vertical, the other linear horizontal. 
EDIT:

If I understand you'd like to have for the polarization what a grating is for the wavelength. I cannot find a passive material that can send to different outputs arbitrary input polarizations (it doesn't mean there is none). So I thought about the following active system (it should work but is quite complicated to do practically): 
You analyze the polarization state and based on the result you actively and automatically do something, like changing the beam direction, if really needed. What you have to do is to measure the Stokes parameters of the input on the fly, and based on that you can steer the beam. In the picture: $X,Y$ are $s$ and $p$ components; $a,b$ are linear components rotated by 45°. Every letter corresponds to an intensity ($|E|^2$).
As you know that the input is linear, you need to find only its orientation $\theta=1/2  \arctan(U/Q)$, where $Q=X-Y$ and $U=a-b$
I'm curious to see if a passive solution exists.
A: The polarizing beam splitter responds to the electric field of the light; the polarization state is defined by the orientation of the electric field with respect to the plane of incidence of a ray of light; if the electric field is parallel to the plane of incidence it is called P polarized; if it is at right angles, sideways, it is called S polarized; these are the first letters of corresponding German words. 
If the electric field is neither S nor P then it is composed of both; there is an S component, and a P component; their vector sum is the electric field.
When only naturally occurring materials were used to make polarizing beam splitters, the separation angles were determined by the crystal structure.  Today the beamsplitter is typically engineered to provide orthogonal beams.  You could use a chain of orthogonal beamsplitters to accomplish a division of the original beam into 6 or more parts, with the angles of your choosing.
The easiest way to proceed is to use a combination of polarizing beam splitters (PBS) and half-wave plates (HWP) that are calibrated for the laser wavelength of the beam. Split the beam, insert a HWP in each output beam, set to rotate the plane of polarization; now insert a PBS in each of these beams: this gives four distinct polarization angles.   
