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(3+1)D topological order and lattice gauge theories can host interesting classes of topological defects of varying codimension, of which the well-known point charges and flux loops are special cases. The complete algebraic structure of these defects defines a higher category, and can be viewed as an emergent higher symmetry. This plays an important role both in the classification of phases of matter and the possible fault-tolerant logical gates in quantum error correcting codes. In this talk I will discuss some of our recent progress in understanding of the properties of invertible defects and higher symmetries from distinct perspectives. I will focus on a class of invertible codimension-2 topological defects, which are referred to as twist strings. I will present a number of general constructions for twist strings in both Abelian and non-Abelian (3+1)D lattice gauge theories, in terms of gauging lower dimensional invertible phases, layer constructions, and condensation defects.