We investigate unusual binary Langmuir monolayers with the same long CH3(CH2)(21) hydrocarbon chains and fluorinated -O-CH2CF3 (FEE) versus nonfluorinated -O-CH2CH3 (EE) hydrophilic heads, whose opposite dipoles assist miscibility, in contrast to the equally oriented polar head dipoles of almost all natural or synthetic amphiphiles that minister to phase separation. Although two-component bulk micelles, lipid bilayers, and monolayers with fluorinated and nonfluorinated chains, which also have opposite dipoles, often show phase separation, we find complete miscibility and nonideality of the FEE-EE mixtures demonstrated via deviation of the composition dependencies of the mean molecular area at fixed surface pressure from the additivity rule. The composition dependencies of the excess molecular areas exhibit minima and maxima which show specific structural changes at particular compositions. They originate from the dipolar and steric interactions between the polar heads, because the interactions between the same chains of FEE and EE do not vary. The pi/A isotherms and the pi/X-FEE phase diagram reveal that mixtures with molar fractions X-FEE >= 0.3 exist in an upright solid phase even in uncompressed state. This result is confirmed by the compressibility values and via Brewster angle microscopy, which does not show optical anisotropy at X-FEE >= 0.3. Comparison of the collapse and phase-transition molecular areas with literature data suggests that the upright architecture corresponds to LS-phase or S-phase with more defects as the S-phase in the pure monolayers. The mixtures with X-FEE < 0.3 exist in tilted L2' phase at low surface pressures. Their mean molecular areas are smaller than the corresponding values in the EE film, which manifests reduction of the tilt of the EE chains with increasing FEE content. We ascribe the chain erection to partial dehydration of the EE heads caused by dipolar attraction between the EE and FEE heads. The excess free energy of mixing Delta G(exc)(pi) is positive but much smaller than the negative total free energy of mixing Delta G(mix)(pi) showing a spontaneous miscibility at all compositions due to an entropy increase. The analysis of the conflict between the Delta G(mix)(pi) minimum at molar fraction X-FEE = 0.5 and the minimum and negative value of the excess molecular area A(pi,exc) at X-FEE = 0.8 shows that the A(pi,exc)/X-FEE minimum has not an electrostatic but a short-range structural origin.