This is confusing because singular use of "were" is a part of the subjunctive mood which sees very little use in day-to-day English speech. As such, most people are unclear on the grammatical rules surrounding it.
(The verb conjugations for the subjunctive mood overlap enough with other moods that people tend to understand it more as a set of special-case exceptions that are followed "just because" rather than as a coherent grammatical entity.)
In a subjunctive sentence, you use the subjunctive forms (eg. "were") in the dependent clause, and the conditional forms (eg. would) in the independent clause.
"If I were to read them, I would have felt that I was cheating."
Without the comma, "I would have felt that I was cheating if I were to read them."
I'll try to demonstrate the separation between a fully conditional sentence and one using the subjunctive mood with an example:
Subjunctive: "If I were rich, I wouldn't be living here." (A hypothetical.)
Conditional: "If I was rich, I wouldn't be living here." (A statement about the past, based on present evidence.)
...or, comparing subjunctive and indicative:
Subjunctive: "Long live the king" or its more obvious form, "May the king live long".
Indicative: "Long lives the king" and its more obvious form "The king lives long".