Polarization is usually defined in terms of the E field direction, i.e.
E_0 in the usual expression for a monochromatic plane wave,
E(x,t) = Re{E_0 exp(i(k dot x - omega*t) }, where x is the position
vector (i.e. the location of the observation point in space) and k is
the propagation vector, also called the spatial frequency vector.
The energy carried by any light beam is proportional to |E|**2.
(Strictly speaking it's proportional to the vector cross product of E
and H, but in any given linear medium, |H| is proportional to |E|.) E
is a vector, so it has to have a direction and a nonzero magnitude at
any instant if there's power flowing. (|E| can be zero if the
polarization is exactly linear, but only for an instant.)
Thermal light has no time-averaged polarization, but that's because it's
very broadband. Sunlight has a polarization vector that changes on the
scale of a femtosecond or two.
Once the light has been put through a polarizer, there's no way to
recover that very rapid variation, but there are various approximate
methods for making "depolarizers".
For displays and other such things, circular polarizers are the most
common sort of depolarizer, because they fix the polaroid sunglass
problem, they're cheap, and that's all that's needed. Note that a
circular polarizer so-called doesn't actually work analogously to a
linear polarizer--it's a film polarizer laminated with a quarter-wave
plate on one side, whereas in order for it to pass one helicity
unaltered, it would need a waveplate on each side.
For display use, you put the waveplate on the outside, facing the
viewer, so that it selects one linear polarization from the LCD as
normal, and then converts it to circular, so that the viewer's
sunglasses don't cause problems.
Other approaches sometimes used in instruments are rotating ground-glass
diffusers, which make the polarization change on the scale of hundreds
of microseconds at best, or Cornu depolarizers, which are waveplates
whose retardation varies rapidly across the field of view. Cornu
depolarizers are mostly used in grating spectrometers, where they can
adequatly homogenize out the polarization dependence of the grating.
They're not much use otherwise.