The steady advance of quantum technologies in recent years has brought their application to quantum chemistry to the forefront of research. Most quantum algorithms for quantum chemistry work by expressing the Hamiltonian in a basis of qubits and fragmenting the Hamiltonian into individual qubit operations. The product of exponentials of these operations then approximate the exponential of the Hamiltonian. However, random orders of fragments generally break symmetries of the Hamiltonian, such as spin and spatial symmetry. Abelian symmetries are local in a single-particle, symmetry-adapted basis and can thus be easily restored by grouping fragments. We show that the same strategy cannot restore non-Abelian symmetries. We demonstate an alternate approach whereby non-commuting terms can be projected out of a symmetry-conserving operator and rotated into a set of commuting terms such that they are exactly exponentiated and symmetry is conserved. Finally, we show the consequences of symmetry conservation for quantum chemistry. Symmetries have an exponential advantage for classical computers relative to quantum computers, enabling large-scale tests of quantum algorithms while the technology develops.