  Giant radio galaxies (GRGs) are physically large radio sources that extend
well beyond their host galaxy environment. Their polarization properties are
affected by the poorly constrained magnetic field that permeates the
intergalactic medium on Mpc scales. A low frequency ($<$ 200 MHz) polarization
study of this class of radio sources is now possible with LOFAR. Here we
investigate the polarization properties and Faraday rotation measure (RM) of a
catalog of GRGs detected in the LoTSS. This is the first low-frequency
polarization study of a large sample of radio galaxies selected on their
physical size. We explore the magneto-ionic properties of their under-dense
environment and probe intergalactic magnetic fields using the Faraday rotation
properties of their radio lobes. We use RM synthesis in the 120-168 MHz band to
search for polarized emission and to derive the RM and fractional polarization
of each detected source component. We study the depolarization between 1.4 GHz
and 144 MHz using images from the NVSS. From a sample of 240 GRGs, we detected
37 sources in polarization, all with a total flux density above 56 mJy. The
fractional polarization of the detected GRGs at 1.4 GHz and 144 MHz is
consistent with a small amount of Faraday depolarization (a Faraday dispersion
$<$ 0.3 rad m$^{-2}$). Our analysis shows that the lobes are expanding into a
low-density ($<10^{-5}$ cm$^{-3}$) local environment permeated by weak magnetic
fields ($<$0.1 $\mu$G) with fluctuations on scales of 3 to 25 kpc. The presence
of foreground galaxy clusters appears to influence the polarization detection
rate up to 2R$_{500}$. In general, this work demonstrates the ability of LOFAR
to quantify the rarefied environments in which these GRGs exist and highlights
them as an excellent statistical sample to use as high precision probes of
magnetic fields in the intergalactic medium and the Milky Way.
