How can teachers reduce student suspicions in discrepancies and irregularities of grading? From experience, many teachers dislike it when students compare grades, but that often shows the discrepancies and irregularities in the grading system. What other methods should teachers implement in order to keep grading transparent and fair? 

Consider the following situations in your answer:


Curved Grades
The possibility of an error in grading, which leads to an imbalance in the grade distribution.


Some teachers immediately assume that just because a student wants to compare papers that he/she is just interested in increasing their grade. This is not the case! Some teachers make mistakes so instead of brushing it off, by assuming that students want to "grade-lawyer" or the like, they should consider the possibility that an error has been made. The goal of this question is to reduce such responses.
Something I used to do (I no longer teach) was to grade in such a way that there should not be an issue if any student compared his/her graded test with any other student in the class. And this certainly happened in my classes (indeed, I sometimes did this myself when I was a student), from the first classes I taught in 1983 to the last classes I taught in 2005. By the way, I taught math. I suspect the methods I describe are much easier to carry out in math than in some other fields, such as literary criticism or philosophy.

Something I started doing after a few years of teaching was to photocopy my solutions/rubric sheet and handing a copy of it back with each student's test. Before this, I often handed out solutions to save class time (and office hour time), and at some point I realized I could save even more of my personal time by simply handing back what I had already hand-written for grading purposes, without bothering to rewrite (or type) it again in a neater form. The solutions I used for grading purposes were often brief, but for rubric purposes I would always include steps that I expected some students to miss, so it actually worked fine with most students when they consulted it to see what they did wrong or how to correctly work the problem. And for those places where I was too brief for a particular student, they tended to consult with their neighbors sitting next to them and together they almost always managed to figure things out, so I tended to only get the less trivial types of questions. As for the rubric, what worked best for me was to treat the rubric as a "work in progress" in the sense that I made grading decisions whenever a certain type of error showed up, rather than trying to anticipate them in advance.

One policy I had and which I often reminded students about was that they should never be afraid of asking me about a question for fear that their grade could decrease. If they saw anything that seemed to be an inconsistency in grading (and this only happened on very rare occasions because I tended to document for myself almost every kind of mistake made as I graded tests), then I wanted them to let me know. If in fact I did make a mistake, then my mistake would never lower their grade, but it could increase other students' grades if my mistake was an inconsistency in how many points were counted off for a certain type of mistake on their part. Also, if I saw on a student's paper brought to me where I made an oversight by not counting off for something incorrect, then I would mark in ink a correction (so the student wouldn't at some later time think what was incorrect was actually correct) and include a comment that this was found after the tests were graded.

One issue I used to often see students and teachers arguing about was points taken off for correct answers that were obtained by not entirely complete work. I learned early on that this can cause trouble when trying to justify your grading to students. The most straightforward way I found to handle this was to simply design problems whose solution requires all the things you feel are important, and of course often reminding students when working problems in class prior to a test what type of work is acceptable and what type of work is not acceptable. For example, the standard elementary calculus method for determining the global maximum and global minimum of a continuous function on a closed and bounded interval involves finding the values of the function at the critical points (where the derivative is zero or undefined) in the interior of the interval and the values of the function at the endpoints of the interval. So if you give a problem where both extrema occur at the endpoints, or both extrema occur in the interior of the interval, then the student could get the correct answer using correct (but not entirely complete) mathematical reasoning by only considering the critical points or by only considering the endpoints. The way to fix this is to design the problem so that one of the extrema is in the interior and the other extrema is at an endpoint. Better still is to arrange it so that there are at least two critical points, with at least one critical point lying outside the interval and at which the function's value is greater than the global maximum on the interval or less than the global minimum on the interval.