The antiferromagnetic Heisenberg chain is expected to have an extended symmetry,  [SU(2)×SU(2)]/Z2, in the infrared limit whose physical interpretation is that the spin and dimer order parameters form the components of a common four-dimensional pseudovector. Here we numerically investigate this emergent symmetry using quantum Monte Carlo simulations of a modified Heisenberg chain (the J/Q model) in which the logarithmic scaling corrections of the conventional Heisenberg chain can be avoided. We show how the two- and three-point spin and dimer correlation functions approach their forms constrained by conformal field theory as the system size increases and numerically confirm the expected effects of the extended symmetry on various correlation functions. We stress that sometimes the leading power laws of three-point (and higher) correlations are not given simply by the scaling dimensions of the lattice operators involved but can be faster decaying because of exact cancellations of contributions from the fields and currents under conformal symmetry.