Iterated cofibers rotate with suspension reflection

The product-pair construction for the well-pointed pair (X,{x0}) and the interval endpoints gives a cofibration of $X\times \mathrm{\partial }I\cup \{x_0\}\times I$ into $X\times I$. Collapse the top and basepoint track: pushout HEP gives the based cone-base cofibration $X\to CX$. Its pushout along f makes $i:Y\to C_f$ a cofibration. F3 therefore gives the homotopy equivalence $\psi :C_i\to C_f\mathrm{/}Y=\mathrm{\Sigma }X$. On the included copy $j(C_f)$ this collapse is literally $q$, so $\psi \circ j=q$.

In $\mathrm{\Sigma }{C}_f$, write a point of the cone portion as (x,s,t), with s the cone coordinate and t the suspension coordinate. Sending it to (x,t,s) in $C(\mathrm{\Sigma }f)$, and [y,t] to [y,t] on the target part, respects every collapsed subset and the attaching relation. The same coordinate interchange is its continuous inverse by F4. The resulting final quotient on ${\mathrm{\Sigma }}^{2}X$ interchanges its two suspension coordinates, rather than acting identically.

Identify $C_i$ with $CX{\cup }_{f}CY$. On CX set $H([x,s],t)=[f(x),t-st]$ in $\mathrm{\Sigma }Y$, and on CY set $H([y,s],t)=[y,s+t-st]$. At s=0 the two values agree via f. At the CX tip s=1 the value is [f(x),0]=; at the CY tip it is [y,1]=. The basepoint tracks are constant. F4 descends these continuous formulas. At t=0 the map collapses CX and is the usual quotient of CY; at t=1 it is $-\mathrm{\Sigma }f$ after ψ on CX and is constant on CY. This proves the claimed reflected next arrow.

To identify the interchange sign explicitly, model $I^2\mathrm{/}\mathrm{\partial }{I}^2$ by the one-point compactification of ${\mathbb{R}}^2$ using the same increasing coordinate homeomorphism in each variable. Swap is the linear map T(v1,v2)=(v2,v1), and first-coordinate reflection is R(v1,v2)=(-v1,v2). The matrix $R^{-1}T$ is a quarter-turn rotation. A path of rotations from it to the identity gives a path of invertible matrices from T to R. These matrices are orthogonal, so they preserve norms uniformly and extend to a homotopy fixing infinity. Keeping X as a parameter and collapsing its basepoint gives interchange homotopic to single-coordinate reflection on ${\mathrm{\Sigma }}^{2}X$. Thus iteration preserves exactly the reflection sign in the cofiber convention.