We report on the discovery of gamma-ray pulsations from five millisecond pulsars (MSPs) using the Fermi Large Area Telescope (LAT) and timing ephemerides provided by various radio observatories. We also present confirmation of the gamma-ray pulsations from a sixth source, PSR J2051−0827. Five of these six MSPs are in binary systems: PSRs J1713+0747, J1741+1351, J1600−3053 and the two black widow binary pulsars PSRs J0610−2100 and J2051−0827. The only isolated MSP is the nearby PSR J1024−0719, which is also known to emit X-rays. We present X-ray observations in the direction of PSRs J1600−3053 and J2051−0827. While PSR J2051−0827 is firmly detected, we can only give upper limits for the X-ray flux of PSR J1600−3053. There are no dedicated X-ray observations available for the other three objects. The MSPs mentioned above, together with most of the MSPs detected by Fermi, are used to put together a sample of 30 gamma-ray MSPs. This sample is used to study the morphology and phase connection of radio and gamma-ray pulse profiles. We show that MSPs with pulsed gamma-ray emission which is phase-aligned with the radio emission present the steepest radio spectra and the largest magnetic fields at the light cylinder among all MSPs. Also, we observe a trend towards very low, or undetectable, radio linear polarization levels. These properties could be attributed to caustic radio emission produced at a range of different altitudes in the magnetosphere. We note that most of these characteristics are also observed in the Crab pulsar, the only other radio pulsar known to exhibit phase-aligned radio and gamma-ray emission.