Heisenberg spin-1/2 chain materials are known to substantially alter their static and dynamic properties when experiencing an effective transverse staggered field originating from the varying local environment of the individual spins. We present a temperature-, angular- and field-dependent ²⁹Si NMR study of the model compound BaCu₂Si₂O₇. The experimental data are interpreted in terms of the divergent low-temperature transverse susceptibility, predicted by theory for spin chains in coexisting longitudinal and transverse staggered fields. First, our analysis employs a finite-temperature density matrix renormalization group (DMRG) study of the relevant one-dimensional Hamiltonian. Next, we compare our numerical with the presently known analytical results. With an analysis based on crystal symmetries, we show how the anisotropic contribution to the sample magnetization is experimentally accessible even below the ordering temperature, in spite of its competition with the collinear order parameter of the antiferromagnetic phase. The modification of static and dynamic properties of the system due to the presence of a local transverse staggered field (LTSF) acting on the one-dimensional spin array are argued to cause the unusual spin reorientation transitions observed in BaCu₂Si₂O₇. On the basis of a Ginzburg-Landau type analysis, we discuss aspects of competing spin structures in the presence of magnetic order and of the enhanced transverse susceptibility.