The column filtration spectral sequence of a first quadrant double complex

Define $D_{a,b}=C_{b,a}$, with $h_{a,b}^D=v_{b,a}^C$ and $v_{a,b}^D=h_{b,a}^C$. The two square-zero identities for $D$ are those for $v^C,h^C$ respectively, and its mixed sum is the mixed sum for $C$ with the two summands exchanged. Thus $D$ is again a first-quadrant anticommuting double complex. In each total degree, permutation of the finite summands gives an isomorphism $\mathrm{Tot}(D)\to T$. It commutes with total differentials because it changes $h^D+v^D$ into $v^C+h^C=h^C+v^C$. The row cutoff $b\le p$ in $D$ becomes the column cutoff in $C$.

Apply the row theorem to $D$. Its horizontal homology at $(q,p)$ is $H_q^v(C_{p,\ast })$, and its induced vertical differential is the original $h^C$. Its second page is therefore $H_p^h(H_q^v(C))$, with the same filtered bidegrees. These are the pages of the column-filtered $T$: the filtration-preserving isomorphism in step 1.1 identifies their graded objects and differentials, and natural page transitions propagate the identification through every page. No sign twist is needed in the transposition.

The same filtered isomorphism sends $H_n(F_p^{\mathrm{r}\mathrm{o}\mathrm{w}}\mathrm{Tot}(D))\to H_n(\mathrm{Tot}(D))$ to $H_n(F_p^{\mathrm{c}\mathrm{o}\mathrm{l}}T)\to H_n(T)$, hence sends their images to each other. Naturality of finite convergence identifies the stationary page in step 2.1 with the stated associated graded of these images. The bounds are $-1$ and $n$ for $n\ge 0$; for negative $n$ the complex is zero. This includes $n=0$, a zero complex and a complex in one column. All maps are specified by finite permutations and universal properties, so no choice assumption enters.