Work and heat exchanger networks have recently drawn increasing attention due to their paramount importance in achieving energy savings. In this work, we introduce a new optimization model for the cost-effective synthesis and energy integration of work and heat exchanger networks considering unclassified process streams (ie., streams whose classification as hot or cold streams cannot be defined a priori). Our innovative modelling approach combines mathematical programming techniques and the pinch location method to obtain an optimal network design with minimal cost, while adjusting pressure and temperature levels of unclassified streams. We propose disjunctive operators for the selection of pressure manipulation equipment, and streams identity classification depending on energy requirements and process operating conditions. In addition, our approach addresses previous shortcomings by eliminating the need for: (i) assigning a specific route of pressure manipulation; and, (ii) classifying streams as low or high-pressure streams; which provides further flexibility to the system. Our methodology is also able to effectively deal with variable inlet and outlet streams temperatures to reach specific optimization goals. The model is solved to global optimality through the minimization of the process total annualized cost. Besides improved computational performance, results from energy analyses reveal that streams classification during process optimization can be greatly advantageous for both subambient and above-ambient applications. In the liquefied natural gas process, it reduces up to 89% the energy demand when compared to literature records.