We provide a complete classification of paradoxical closed-loop $n$-linkages, where $n\geq6$, of mobility $n-4$ or higher, containing revolute, prismatic or helical joints. We also explicitly write down strong necessary conditions for $nR$-linkages of mobility $n-5$. Our main new tool is a geometric relation between a linkage $L$ and another linkage $L'$ resulting from adding equations to the configuration space of $L$. We then lift known classification results for $L'$ to $L$ using this relation.