Economic growth results from countries' accumulation of organizational and technological capabilities. The Economic and Product Complexity Indices, introduced as an attempt to measure these capabilities from a country's basket of exported products, have become popular to study economic development, the geography of innovation, and industrial policies. Despite this reception, the interpretation of these indicators proved difficult. Although the original Method of Reflections suggested a direct interconnection between country and product metrics, it has been proved that the Economic and Product Complexity Indices result from a spectral clustering algorithm that separately groups similar countries or similar products, respectively. This recent approach to economic and product complexity conflicts with the original one and treats separately countries and products. However, building on previous interpretations of the indices and the recent evolution in spectral clustering, we show that these indices simultaneously identify two co-clusters of similar countries and products. This viewpoint reconciles the spectral clustering interpretation of the indices with the original Method of Reflections interpretation. By proving the often neglected intimate relationship between country and product complexity, this approach emphasizes the role of a selected set of products in determining economic development while extending the range of applications of these indicators in economics.