How does the statement of purpose of students help professors to write letters of recommendation for graduate school application in U.S.? Usually, professors need to see students' statement of purpose before they write letters of recommendation for them. However,

(1) The most important part of LoRs is the comparison of the applicant with their classmates/peer, which is normally not shown in the statement of purpose.

(2) Professors often talk about students' strengths in LoRs, but they are not supposed to use the strengths mentioned(or boasted of) in the students' statement of purpose when the strengths are unknown to professors before they see the statement of purpose.

(3) Professors may want to know students' experiences, interests and plans for future. But such things are already mentioned in students' statement of purpose and they don't have to repeat them again in the letters of recommendation(professors are not supposed to know students experiences, interests and plans for future better than students themselves, right?).

So my question is, given (1), (2) and (3), why do professors need to look at students' statement of purpose at all when they write LoRs for them? How does the statement of purpose help professors to write letters of recommendation?
The statement of purpose is helpful to me in writing a letter of recommendation for a student because I can say things in the letter that relate to the specific plans that the student has beyond the more general plan of entering a graduate program.  

For example, a student might ask me to write a letter of recommendation for admission to a PhD program in applied mathematics at University X.  Knowing that the student plans to specialize in a subfield (say computational optimization) gives me the chance to write into the letter that the student is well prepared for graduate study in this area because of (whatever factors might be relevant, including specific courses taken, programming languages the student knows, etc.)  Those same experiences might not be relevant if the student wants to work in a different subfield.

Another reason for me to want to see the student's statement of purpose is so that I can provide helpful advice to the student.  Many students are naive and prepare draft statements that won't help and might actually hurt their chances of admission.  If I've read the draft statement I can suggest changes.  If it's really bad, then I might decide that I'm unwilling to write a recommendation letter for the student because it would reflect badly on me.  

Ultimately it really doesn't matter why the recommender wants to see your statement- you're not in any position to demand a letter of recommendation, so you should provide whatever information the requestor asks you for.