Standard Gini covariance and Gini correlation play important roles in measuring the dependence of random variables with heavy tails. However, the asymmetry brings a substantial difficulty in interpretation. In this paper, we propose a symmetric Gini-type covariance and a symmetric Gini correlation (\(\rho_g\)) based on the joint rank function. The proposed correlation \(\rho_g\) is more robust than the Pearson correlation but less robust than the Kendall's \(\tau\) correlation. We establish the relationship between \(\rho_g\) and the linear correlation \(\rho\) for a class of random vectors in the family of elliptical distributions, which allows us to estimate \(\rho\) based on estimation of \(\rho_g\). The asymptotic normality of the resulting estimators of \(\rho\) are studied through two approaches: one from influence function and the other from U-statistics and the delta method. We compare asymptotic efficiencies of linear correlation estimators based on the symmetric Gini, regular Gini, Pearson and Kendall's \(\tau\) under various distributions. In addition to reasonably balancing between robustness and efficiency, the proposed measure \(\rho_g\) demonstrates superior finite sample performance, which makes it attractive in applications.