We present an analytical model for the theoretical analysis of spin dynamics and spontaneous symmetry breaking in a spinor Bose-Einstein condensate (BEC). This allows for an excellent intuitive understanding of the processes and provides good quantitative agreement with the experimental results of Scherer et al. [Phys. Rev. Lett. 105, 135302 (2010)]. It is shown that the dynamics of a spinor BEC initially prepared in an unstable Zeeman state mF=0 (|0⟩) can be understood by approximating the effective trapping potential for the state |±1⟩ with a cylindrical box potential. The resonances in the creation efficiency of these atom pairs can be traced back to excitation modes of this confinement. The understanding of these excitation modes allows for a detailed characterization of the symmetry-breaking mechanism, showing how a twofold spontaneous breaking of spatial and spin symmetry can occur. In addition, a detailed account of the experimental methods for the preparation and analysis of spinor quantum gases is given. © 2013 American Physical Society.