My question about sign swapping is explained by this blog

https://www.cairographics.org/cookbook/matrix_conventions/

The reason behind this is called pre-multiplication vs post-multiplication technique of matrix transformation.

Post-multiplication

- transformation matrix appears after the position vector matrix

- position vector matrix in homogenous coordinates is represented as a row matrix.

Pre-multiplication

- transformation matrix appears before the position vector matrix

- here the position vectors in homogenous coordinates are represented in column matrix.

Textbook reference: Mathematical elements for computer graphics Rogers/Adams