The crystal structure of the low-temperature form of AgCuPO4 (i.e., alpha-AgCuPO4) was determined by powder X-ray diffraction and was compared with that of the high-temperature form of AgCuPO4 (i.e., beta-AgCuPO4). The magnetic properties of the two forms were examined by measuring their magnetic susceptibilities and evaluating the relative strengths of their spin-exchange interactions on the basis of spin-dimer analysis. Both forms of AgCuPO4 have layers of Cu2P2O8 alternating with silver-atom double layers; beta-AgCuPO4 has two Cu2P2O8 layers per unit cell, while alpha-AgCuPO4 has one. The coordinate environment of each Cu2+ ion is close to being a distorted square pyramid in alpha-AgCuPO4, but it is close to being a distorted trigonal bipyramid in beta-AgCuPO4. The magnetic susceptibilities of alpha- and beta-AgCuPO4 are well simulated by an antiferromagnetic alternating-chain model, which leads to J/k(B) = -146.1 K and alphaJ/k(B) = -75.8 K for alpha-AgCuPO4, and J/k(B) = -82.6 K and alphaJ/k(B) = -31.7 K for beta-AgCuPO4 (with the convention in which the spin-exchange parameter between two adjacent spin sites is written as 2J). The spin gaps, delta/k(B), obtained from these parameters are 93.7 K for alpha-AgCuPO4 and 62.3 K for beta-AgCuPO4. The strongest spin exchange in both forms of AgCuPO4 comes from a super-superexchange path, and this interaction is stronger for alpha-AgCuPO4 than for beta-AgCuPO4 by a factor of approximately 2, in good agreement with the experiment. Our analysis supports the use of this model for beta-AgCuPO4 and indicates that the spin lattice of alpha-AgCuPO4 would be better described by a two-dimensional net made up of weakly interacting alternating chains.