Holonomy convention

A positive meridian increases the lifted angular coordinate by $L$. Its holonomy will mean the ordered product of affine coordinate transitions encountered along the positively oriented loop, with the last transition acting on the left.

With this convention, crossing the upper seam into the lower seam contributes $J$. The increasing-angle developing map on the universal cover satisfies the inverse equivariance relation

There is no sign contradiction: deck equivariance and the ordered coordinate-transition product use inverse conventions.

The affine reference coordinates retain their origin on $\mathcal{W}$. A basepoint for a loop does not mean that the affine coordinates are re-centred at that basepoint.