We develop a method to identify cosmic voids from the matter density field by adopting a physically-motivated concept that voids are the counterpart of massive clusters. To prove the concept we use a pair of $Lambda$CDM simulations, a reference and its initial density-inverted mirror simulation, and study the relation between the effective size of voids and the mass of corresponding clusters. Galaxy cluster-scale dark matter halos are identified in the Mirror simulation at $z=0$ by linking dark matter particles. The void corresponding to each cluster is defined in the Reference simulation as the region occupied by the member particles of the cluster. We study the voids corresponding to the halos more massive than $10^{13}h^{-1}M_{odot}$. We find a power-law scaling relation between the void size and the corresponding cluster mass. Voids with corresponding cluster mass above $10^{15}h^{-1}M_{odot}$ occupy $sim1%$ of the total simulated volume, whereas this fraction increases to $sim54%$ for voids with corresponding cluster mass above $10^{13}h^{-1}M_{odot}$. It is also found that the density profile of the identified voids follows a universal functional form. Based on these findings, we propose a method to identify cluster-counterpart voids directly from the matter density field without their mirror information by utilizing three parameters such as the smoothing scale, density threshold, and minimum core fraction. We recover voids corresponding to clusters more massive than $3times10^{14}h^{-1}M_{odot}$ at 70--74 % level of completeness and reliability. Our results suggest that we are able to identify voids in a way to associate them with clusters of a particular mass-scale.
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