Can one detect from earth that angular momenta of its axial rotation and its orbital rotation are "almost parallel"? I mean that the angle between these angular momenta is acute as opposed to obtuse (23.5 and not 180-23.5), so that the rotations are "roughly in the same direction". 

Related but distinct question -- when was this actually discovered?
Certainly! Ancient astronomers, eg in Babylon ~3000 years ago were sufficiently familiar with the geometry of the celestial equator and the ecliptic to be able to predict eclipses, and discover the eclipse series now known as the Saros series. But of course, in those days astronomers (mostly) didn't conceive of them in terms of the motion of the Earth. 

OTOH, there were a few ancient astronomers in ancient Greece (notably Aristarchus of Samos), and even in Babylon (Seleucus of Seleucia), who did believe in a heliocentric system. 

The celestial pole is easy to discover, (especially if you're in the northern hemisphere), and it doesn't take a lot of observation to notice that the ecliptic differs from the celestial equator. 

Although it's not possible to observe directly that the Sun's path is the ecliptic, it was known that all of the other celestial wanderers stay fairly close to the ecliptic, and that eclipses only occur when the Moon is very close to the ecliptic. And you can get a pretty accurate idea of the Sun's path by observing the rising &amp; setting of stars and planets that are near the Sun's path. So astronomers across the ancient world recognised that the ecliptic is the Sun's path. And by the 2nd century BC, knowledge of these movements was sufficiently detailed that Hipparchus of Nicea was able to estimate the precession of the equinoxes.