We propose an inverse optimization-based methodology to determine market structure from commodity and transportation prices. The methods are appropriate for locational marginal price-based electricity markets where prices are shadow prices in the centralized optimization used to clear the market. We apply the inverse optimization methodology to outcome data from the Midcontinent ISO electricity market (MISO) and, under noise-free assumptions, recover parameters of transmission and related constraints that are not revealed to market participants but explain the price variation. We demonstrate and evaluate analytical uses of the recovered structure including reconstruction of the pricing mechanism and investigations of locational market power through the transmission constrained residual demand derivative. Prices generated from the reconstructed mechanism are highly correlated to actual MISO prices under a wide variety of market conditions. In a case study, the residual demand derivative is shown to be correlated with coefficients of certain transmission constraints.