Using IMP, Voyager, and Pioneer spacecraft data, we have derived radial intensity profiles of cosmic rays in the heliosphere out to ~70 AU as fractions of the expected interstellar intensities for five particle energies/species during the minimum solar modulation periods of 1987 and 1997. These radial profiles exhibit a markedly different behavior in the two periods of opposite solar magnetic polarity. In the negative polarity period (1987) the radial profiles for the highest-rigidity particles extrapolate to the interstellar (IS) intensity at approximately the estimated location of heliospheric termination shock (TS) at 85--90 AU. The lower-rigidity particle intensities extrapolate to only 0.45--0.60 of the IS intensity at this same location, indicating that a large part of the residual solar modulation for these particles at this time takes place at or beyond the location of the TS. In the 1997 positive polarity time period the fraction of the overall modulation that is occurring between the Earth and the TS is now much smaller, and since the intensity at the Earth is the same in both cycles, the consequence is that the extrapolated radial intensities at the TS location are now only 0.50 (at the highest rigidities) to 0.17 (at the lowest rigidifies) of the respective IS intensities. Thus, at this time of minimum modulation most of the residual modulation apparently is taking place near to or beyond the TS. Also, the ratio of the fraction of the total residual modulation that is taking place between the Earth and the TS and that taking place beyond the TS changes dramatically between positive and negative modulation cycles. Viewed in this new way, contemporary modulation models including drifts, when they are assumed to occur within the TS, do not provide a satisfactory quantitative explanation of the modulation that is taking place at the minimum of solar activity. These models do not predict the large amount of modulation taking place near or beyond the termination shock. This outer heliospheric modulation dominates the overall residual modulation at low energies and is more effective in positive solar magnetic polarity periods. A new paradigm seems to be needed to interpret this large modulation that is taking place in the outer heliosphere. ¿ 2001 American Geophysical Union