Chiral assemblies of nanoparticles (NPs) are typically constructed with helical or tetrahedral geometries. Simple pairs of NPs are not expected to display chirality due to basic symmetry considerations made under the assumption of their spherical geometry. In this study we demonstrate that assemblies consisting of two metallic NPs do possess chirality and strongly rotate polarization of light. Their chiroplasmonic properties are attributed to the prolate geometry of individual colloidal particles. When bridged by biomolecules, the NP pairs acquire scissor-like geometry, with the long axes of NPs forming an angle of ~9°. This small dihedral angle results in chirality of the NP pair, while the consistency of its sign due to the specific conformation of the bridging biomacromolecules breaks the enantiomeric equivalence of the NP pairs. Strong polarization rotation in these nanoassemblies makes possible their utilization in biological analysis. Heterodimers of gold and silver NPs were made using antibody-antigen bridges. Taking advantage of their chiroplasmonic properties, we investigated their bioanalitical potential for detection of an environmental toxin, microcystin-LR, and a cancer biomarker, prostate-specific antigen. The order-of-magnitude improvements in limits of detection compared to all other analytical techniques are attributed to plasmonic enhancement of intrinsic chirality of biological compounds, strong optical coupling of photons with NP assemblies with twisted geometries, and signal amplification due to the bisignate nature of circular dichroism bands.