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Nowadays, the data collected in physical/engineering systems allows various machine learning methods to conduct system monitoring and control, when the physical knowledge on the system edge is limited and challenging to recover completely. Solving such problems typically requires identifying forward system mapping rules, from system states to the output measurements. However, the forward system identification based on digital twin can hardly provide complete monitoring functions, such as state estimation, e.g., to infer the states from measurements. While one can directly learn the inverse mapping rule, it is more desirable to re-utilize the forward digital twin since it is relatively easy to embed physical law there to regularize the inverse process and avoid overfitting. For this purpose, this paper proposes an invertible learning structure based on designing parallel paths in structural neural networks with basis functionals and embedding virtual storage variables for information preservation. For such a two-way digital twin modeling, there is an additional challenge of multiple solutions for system inverse, which contradict the reality of one feasible solution for the current system. To avoid ambiguous inverse, the proposed model maximizes the physical likelihood to contract the original solution space, leading to the unique system operation status of interest. We validate the proposed method on various physical system monitoring tasks and scenarios, such as inverse kinematics problems, power system state estimation, etc. Furthermore, by building a perfect match of a forward-inverse pair, the proposed method obtains accurate and computation-efficient inverse predictions, given observations. Finally, the forward physical interpretation and small prediction errors guarantee the explainability of the invertible structure, compared to standard learning methods.