  Aim: We aim to refine the orbital and physical parameters and determine the
sky-projected planet orbital obliquity of five eccentric transiting planetary
systems: HAT-P-15, HAT-P-17, HAT-P-21, HAT-P-26, and HAT-P-29. Each of the
systems hosts a hot Jupiter, except for HAT-P-26 which hosts a Neptune-mass
planet. Methods: We observed transit events of these planets with the HARPS-N
spectrograph, obtaining high-precision radial velocity measurements that allow
us to measure the Rossiter-McLaughlin effect for each of the target systems. We
used these new HARPS-N spectra and archival data, including those from Gaia, to
better characterise the stellar atmospheric parameters. The photometric
parameters for four of the hot Jupiters were recalculated using 17 new transit
light curves, obtained with an array of medium-class telescopes, and data from
the TESS space telescope. HATNet time-series photometric data were checked for
the signatures of rotation periods of the target stars and their spin axis
inclination. Results: From the analysis of the Rossiter-McLaughlin effect, we
derived a sky-projected obliquity of 13, -26.3, -0.7, -26 degree for HAT-P-15b,
HAT-P-17b, HAT-P-21b and HAT-P-29b, respectively. Due to the quality of the
data, we were not able to well constrain the sky-projected obliquity for
HAT-P-26b, although a prograde orbit is favoured. The stellar activity of
HAT-P-21 indicates a rotation period of 15.88 days, which allowed us to
determine its true misalignment angle (25 degree). Our new analysis of the
physical parameters of the five exoplanetary systems returned values compatible
with those existing in the literature. Using TESS and the available transit
light curves, we reviewed the orbital ephemeris for the five systems and
confirmed that the HAT-P-26 system shows transit timing variations, which may
tentatively be attributed to the presence of a third body.
