The short answer is that both Linear and Circular Polarizers do the same thing. The actual polarization effects such as reducing reflections on glass surfaces, increasing color saturation in foliage, darkening a blue sky are the same with both Linear and Circular polarizers.

Circular Polarizers contains a Linear Polarizer component that does the main work of polarization, as well as a second layer inside the filter called a Quarter Wave Plate, which “spins” the light after it goes through the linear layer and before it enters the camera lens.

The Problem:

The main problem that the circular polarizer addresses is cross polarization on other reflective surfaces in your system such as mirrors and beam splitters. Reflective surfaces polarize light . . . which is why a polarizer can reduce or eliminate those reflections.  If you have a mirror or other reflective surface inside your camera, a linear polarizer can cross polarize the reflected image and possibly black out the image. 

The classic problem in 35mm film production was the video tap. Using a linear polarizer on a film camera with a video tap, could and often would cause the video feed to go dark because the partially silvered mirror in the video tap would get cross polarized.  A very similar cross polarization is used to create Variable Neutral Density Filters with ND values between ND 0.6 (2 Stops) and ND 2.4 (8 stops).  8 stops means that only about 0.39% of the light gets past the filter.  
A Neutral Density Filter with an ND Value of ND 2.4 can be used to allow opening the aperture in a lens for shallow depth of field, or slow a shutter for various effects . . . it is not a good thing to do accidentally in an optical system with a mirror.

In still photography, SLR cameras and DSLR cameras have a mirror, which a linear polarizer can cross polarize, causing the image to darken or go black.

The Circular Pol Solution to Cross Polarization:

With a circular pol, the quarter wave plate on the rear of the polarizer spins the light before it enters the camera lens so that it doesn't get cross polarized on any reflective surfaces in the system, such as  the partial mirror in a video tap or a DSLR mirror.

If there are mirrors in your optical system, the circular pol solves any problems or potential problems.

The linear polarizer element in a circular polarizer needs to be out front, pointed at the world, with the quarter wave plate on the rear side, the camera lens side. A circular polarizer doesn’t work if you get it in backwards. With a screw-in filter that’s no issue, there's only one way you can screw it on the lens.. 

Polarizers used in cinematography are often inserted in a matte box so there's an opportunity to insert a circular pol backwards and it doesn't polarize when inserted incorrect. 

With drop-in filters, rectangular filters and Rota-Pols for cine matte boxes, circular polarizing filters are labeled with “this side out”.

Circular polarizers for cinematography are marked to show proper orientation, but mistakes happen.   If you have no mirrors or reflective surfaces in your system, such as a mirror or a beam splitter . . . then the Linear-Pol will not cause you any trouble and you can’t put it in the matte box backwards, which might save you some time and trouble.

So if you are asking the question:  "Do I need a circular or linear polarizer?", we hope we've given you enough information to make that decision.

Lindsey Optics offers both circular and linear polarizers in professional cine sizes in 4.5" Round Drop-In, 138mm Round Drop-In, and 4"x5.65" and Rota-Pols in 4x5.65" and 6.6x6.6" all with anti-reflection coating.

Our Rota-Pol a circular polarizer made to fit in 4"x5.65" cine matte boxes and allow quick and easy rotation by hand or with a motor: