The mechanism of high-temperature superconductivity in the newly discovered iron-based superconductors is unresolved. We use spectroscopic imaging-scanning tunneling microscopy to study the electronic structure of a representative compound CaFe1.94Co0.06As2 in the "parent" state from which this superconductivity emerges. Static, unidirectional electronic nanostructures of dimension eight times the inter-iron-atom distance a(Fe-Fe) and aligned along the crystal a axis are observed. In contrast, the delocalized electronic states detectable by quasiparticle interference imaging are dispersive along the b axis only and are consistent with a nematic alpha2 band with an apparent band folding having wave vector q vector congruent with +/-2pi/8a(Fe-Fe) along the a axis. All these effects rotate through 90 degrees at orthorhombic twin boundaries, indicating that they are bulk properties. As none of these phenomena are expected merely due to crystal symmetry, underdoped ferropnictides may exhibit a more complex electronic nematic state than originally expected.