Can a teacher to refuse to follow their late work policy? My teacher handed out a syllabus at the beginning of the year,  claiming to have a late work policy where he only reduces the max percent you can get on an assignment for every day it's late. The exact writing of his late work policy can be found here.

Two weeks into the school year, I handed in an assignment 1 day late, and received half credit. I assumed I received half credit because we went over it in class, so I didn't think anything of it. Three weeks later, I handed in another assignment 1 day late, which I received half credit for too. 

When I confronted my teacher, his explanation was that he added the exact writing "The teacher reserves the right to change or amend this syllabus at any time without notice." inside of the syllabus.

Regardless of it not being morally right, is a teacher allowed to do this? I see no reason to give students a late work policy and then completely disregard it. Because I was unaware of him ignoring the policy, my grade dropped by nearly 20%(and my school is already half way through the first term).

Are teachers/professors allowed to change to completely ignore a late-work policy without notice?
I doubt anyone can state conclusively that it is wrong for a teacher to change their policies in the middle of the semester. However, I think many can agree that a teacher does not want to be perceived as whimsical or mercurial by their students. If a teacher develops a habit of arbitrarily changing policies in the middle of a course to the detriment of students it can create a lot of ill-will between the teacher and the students. 

Sometimes it is necessary to make changes. Examples include a true oversight by the teacher that needs to be addressed, obvious abuses by students of a policy, etc. Becoming more lenient is always popular while become more stringent should be in response to inappropriate behavior.

Whenever a change is necessary it must be communicated to the class clearly with some form of explanation. This is as a courtesy to maintain a productive learning environment. If this is neglected students will search for ways to fight against what they believe is unfair changes through such as methods as complaining, low course evaluations, or even academic dishonesty.