  Expanding on our prior efforts to search for Lorentz invariance violation
(LIV) using linear optical polarimetry of extragalactic objects, we propose a
new method that combines linear and circular polarization measurements. While
existing work focused on the tendency of LIV to reduce the linear polarization
degree, this new method additionally takes into account the coupling between
photon helicities induced by some models. This coupling can generate circular
polarization as light propagates, even if there is no circular polarization at
the source. Combining significant detections of linear polarization of light
from extragalactic objects with the absence of a detection of circular
polarization in most measurements results in significantly tighter constraints
on LIV. The analysis is carried out in the framework of the Standard-Model
Extension (SME), an effective field theory framework to describe low-energy
effects of an underlying more fundamental quantum gravity theory. We evaluate
the performance of our method by deriving constraints on mass dimension $d=4$
CPT-even SME coefficients from a small set of archival circular and linear
optical polarimetry and compare them to similar constraints derived in previous
work with far larger sample sizes and based on linear polarimetry only. The new
method yields constraints that are an order of magnitude tighter even for our
modest sample size of 21 objects. Based on the demonstrated gain in
constraining power from scarce circular data, we advocate for the need in
future extragalactic circular polarization surveys.
