From many data types, we hope identify the phylogenetic semidirected network that generated the data, but not the rooted network. In the semidirected network, some but not all edges are directed. I will define a general class of semidirected phylogenetic networks, with a stable set of leaves, tree nodes and hybrid nodes. This class includes both rooted phylogenetic trees and networks, and unrooted phylogenetic trees. I will show various fundamental properties for these networks, including how to generalize the ""tree-child"" property from rooted to semidirected networks. I then will define an edge-based representation of semidirected phylogenetic networks, which generalizes the node-based μ-representation of a rooted network by Cardona et al. (2009), and the split-based representation of an unrooted tree. It leads to a dissimilarity between semidirected networks, which can be efficiently computed in near-quadratic time, and extends the widely-used Robinson-Foulds distance on both rooted trees and unrooted trees. This dissimilarity is in fact a distance on the space of tree-child semidirected phylogenetic networks.