  We present general analytical criteria for the precise design of symmetric or
"antimetric" 2-port systems, including ideal standard filters (Chebyshev,
elliptic, etc.), based on the non-normalized resonant (quasi-normal) modes of
the system. We first develop a quasi-normal mode theory (QNMT) to calculate a
system's scattering $S$ matrix, satisfying both energy conservation and
reciprocity, and describe how low-$Q$ modes can be combined into an effective
slowly varying background response $C$. We then show that the resonant QNM
fields of all lossless reciprocal 2-port systems with a symmetric
($S_{22}=S_{11}$) or "antimetric" ($S_{22}=-S_{11}$) response couple to the
input and output ports with specific unitary ratios, whose relative signs
determine the position of the scattering zeros. This allows us to obtain design
criteria assigning values to the poles, background response, and QNM-to-ports
coupling coefficients. Filter devices can then be designed via a
well-conditioned nonlinear optimization (or root-finding) problem using a
numerical eigensolver. As an example application, we design microwave
metasurface filters that precisely match standard amplitude filters of various
orders and bandwidths, with focus on the best-performing elliptic filters.
