One of the main procedures used in modern chemistry in order to synthetize large molecules with complex architectures has been the assembly, either in a sequential or concerted process, of smaller building blocks. The success of this type of synthetic procedures has lead to completely new fields of chemistry such as that of hybrid organic-inorganic compounds or metal-organic frameworks that have received a great attention in the last years mainly due to their complex properties inherited in part from their building blocks. From the point of view of symmetry, this modular assembling procedures lead sometimes to amazingly complex structures with an overall low symmetry in which local fragments reminiscent of the different building blocks retain practically all of their symmetry. In this communication we will show how the formalism of continuous symmetry measures may be efficiently applied to analyze to which degree different sets of atoms retain local symmetries when assembled in a larger molecule or a supramolecular system with a different global symmetry. Using selected examples from different fields of chemistry, encompassing inorganic, organometallic, organic or even biological chemistry we will illustrate some of the most relevant consequences of this interplay between local and global symmetries in molecular chemistry. We will show that continuous symmetry measures, besides providing an elegant way of describing molecular symmetry in a way complementary to the classical group theoretical approach, allow a deeper understanding of symmetry and its consequences in modular structures obtained from the assembly of prebuild fragments. Besides the obvious geometrical symmetry of the nuclear arrangement, we will show that continuous symmetry measures can be successfully applied to the analysis of other aspects of molecules especially relevant to chemistry such as the electron density, where the remaining local symmetry of the individual building blocks may influence its global features. Special emphasis will be made for systems with incompatible local symmetries, for example collinear rotation axes of incommensurate order in sandwich complexes such as (C6H6)M(C5H5), showing how the local symmetry of one part of the molecule progressively changes till it is converted to a different local symmetry.