  We study the long-standing open problem of efficiently testing rectilinear
planarity of series-parallel graphs (SP-graphs) in the variable embedding
setting. A key ingredient behind the design of a linear-time testing algorithm
for SP-graphs of vertex-degree at most three is that one can restrict the
attention to a constant number of ``rectilinear shapes'' for each series or
parallel component. To formally describe these shapes the notion of spirality
can be used. This key ingredient no longer holds for SP-graphs with vertices of
degree four, as we prove a logarithmic lower bound on the spirality of their
components. The bound holds even for the independent-parallel SP-graphs, in
which no two parallel components share a pole. Nonetheless, by studying the
spirality properties of the independent-parallel SP-graphs, we are able to
design a linear-time rectilinear planarity testing algorithm for this graph
family.
