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Mathematica, 33 bytes

Differences[#,1,2]~MatchQ~{0...}&

Unnamed function taking a list of numbers (or even more general objects) and returning True or False. The function Differences[#,1,2] takes the differences, not of consecutive pairs of integers, but pairs of integers at distance two apart. Then we just check whether the resulting list has nothing other than zeros in it.

As a bonus, for one more byte (change the 2 to #2), we get a function that inputs a list of integers and another positive integer #2, and checks whether the input list is the result of interleaving #2 constant sequences periodically with one another. For example,

Differences[#,1,#2]~MatchQ~{0...}&[{1,2,3,4,5,1,2,3,4,5,1,2},5]

evaluates to True.