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 .