We consider the asymptotic joint distributions among several families of well-known metrics on $S_n$, the symmetric group. These include the bi-invariant metrics such as the Cayley and Hamming distance, and the left-invariant metrics such as Spearman's footrule, Kendall's tau, and the Ulam distance. We also introduce a natural limit of the Spearman family, $ρ_\infty$, and study its asymptotic distribution and relation with other metrics. This is a continuation of earlier work on the asymptotic independence of bi-invariant metrics on both $S_n$ and general linear groups over a finite field. The technique is based on some simple observation about the record map and Hammersley's device. In several cases, we give near-optimal estimate of the error term for asymptotic independence. This simplifies significantly the proof of a central limit theorem by Bai, Chao, and Liang regarding the oscillation of a permutation.