We have completed part of a program to study the X-ray emission properties of old rotation-powered pulsars with XMM-Newton in order to probe and identify the origin of their X-radiation. The X-ray emission from these old pulsars is largely dominated by nonthermal processes. None of the observed spectra required adding a thermal component consisting of either a hot polar cap or surface cooling emission to model the data. The X-ray spectrum of PSR B0950+08 is best described by a single power law of photon index α = 1.93. Taking optical data from the VLT FORS1 into account, a broken power-law model with the break point E_break = 0.67 keV and the photon indices α₁ = 1.27 and α₂ = 1.88 for E < E_break and E > E_break, respectively, is found to describe the pulsar's broadband spectrum from the optical to the X-ray band. Three σ temperature upper limits for possible contributions from a heated polar cap or the whole neutron star surface are T < 0.87 × 10⁶ and T < 0.48 × 10⁶ K, respectively. We also find that the X-ray emission from PSR B0950+08 is pulsed with two peaks per rotation period. The phase separation between the two X-ray peaks is ~144° (maximum to maximum), which is similar to the pulse peak separation observed in the radio band at 1.4 GHz. The main radio peak and the trailing X-ray peak are almost phase-aligned. The fraction of X-ray pulsed photons is ~30%. A phase-resolved spectral analysis confirms the nonthermal nature of the pulsed emission and finds no spectral variations as a function of pulse phase. Detailed pulse profile simulations using the polar gap, the outer gap, and the two-pole caustic model constrain the pulsar's emission geometry to be that of an almost orthogonal rotator, for which the two-pole caustic model can reproduce the observed doubly peaked X-ray pulse profile. The spectral emission properties observed for PSR B0823+26 are similar to those of PSR B0950+08. Its energy spectrum is very well described by a single power law with photon index α = 2.5. Three σ temperature upper limits for thermal contributions from a hot polar cap or from the entire neutron star surface are T < 1.17 × 10⁶ and T < 0.5 × 10⁶ K, respectively. There is evidence for pulsed X-ray emission at the ~97% confidence level with a pulsed fraction of 49% ± 22%. For PSR J2043+2740, which is located ~1° outside the boundary of the Cygnus Loop, we report the first detection of X-ray emission. A power-law spectrum or a combination of a thermal and a power-law spectrum both yield acceptable descriptions of its X-ray spectrum. No X-ray pulses are detected from PSR J2043+2740 and the sensitivity is low—the 2 σ pulsed fraction upper limit is 57% assuming a sinusoidal pulse profile.