  In this paper we present new empirical radio surface brightness-to-diameter
({\Sigma} - D) relations for supernova remnants (SNRs) in our Galaxy. We also
present new theoretical derivations of the {\Sigma} - D relation based on
equipartition or on constant ratio between cosmic rays and magnetic field
energy. A new calibration sample of 60 Galactic SNRs with independently
determined distances is created. Instead of (standard) vertical regression,
used in previous papers, different fitting procedures are applied to the
calibration sample in the log {\Sigma} - log D plane. Non-standard regressions
are used to satisfy the requirement that values of parameters obtained from the
fitting of {\Sigma} - D and D - {\Sigma} relations should be invariant within
estimated uncertainties. We impose symmetry between {\Sigma} - D and D -
{\Sigma} due to the existence of large scatter in both D and {\Sigma}. Using
four fitting methods which treat {\Sigma} and D symmetrically, different
{\Sigma} - D slopes {\beta} are obtained for the calibration sample. Monte
Carlo simulations verify that the slopes of the empirical {\Sigma} - D relation
should be determined by using orthogonal regression, because of its good
performance for data sets with severe scatter. The slope derived here ({\beta}
= 4.8) is significantly steeper than those derived in previous studies. This
new slope is closer to the updated theoretically predicted surface
brightness-diameter slope in the radio range for the Sedov phase. We also
analyze the empirical {\Sigma} - D relations for SNRs in the dense environment
of molecular clouds and for SNRs evolving in lower-density interstellar medium.
Applying the new empirical relation to estimate distances of Galactic SNRs
results in a dramatically changed distance scale.
