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INFN is developing at the LASA lab (Milano, Italy) the High Order (HO) corrector magnets for the High Luminosity-LHC (HL-LHC) project, which will equip the interaction regions. Five prototypes, from skew quadrupole to dodecapole, have been designed and tested at LASA. All the HO correctors are based on a superferric design, which allows a relatively simple, modular, and easy to construct magnet. This modularity has been exploited for an engineering change request. The tradeoff between safe handling, force management during cooldown, powering and protection of the magnet is explained in detail. The design of the coils and the mechanical assembly have been improved to increase the efficiency of the series production. Most of the optimizations are aimed to guarantee both a good integration in the LHC framework (i.e., radiation hardness, easiness of installation, safe operation over years), and compatibility with the series production (i.e., standardization of procedures and components for different magnets, reduction of the number of components, standardization of the quality control systems and acceptance tests). All of the produced magnets will be tested at LASA and then delivered to CERN. The design of the test-bed and the optimization of cryogenic and testing procedures is also described in this paper. Following the completion of the prototyping phase, we report in detail the solutions chosen for the HO correctors and we give a perspective of the series construction in industry and acceptance tests at LASA.