We measured angular distributions of recoil-polarization response functions for neutral pion electroproduction for W=1.23  GeV at Q2=1.0  (GeV/c)2, obtaining 14 separated response functions plus 2 Rosenbluth combinations; of these, 12 have been observed for the first time. Dynamical models do not describe quantities governed by imaginary parts of interference products well, indicating the need for adjusting magnitudes and phases for nonresonant amplitudes. We performed a nearly model-independent multipole analysis and obtained values for Re (S1+/M1+)=-(6.84±0.15)% and Re (E1+/M1+)=-(2.91±0.19)% that are distinctly different from those from the traditional Legendre analysis based upon M1+ dominance and ℓπ≤1 truncation.