K (ngn/k), 22 bytes

Returns odd output before even output.

2/64 2#,/|@\^:\(64#2)\

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(64#2)\ Get an array of all bits of the integer.
,/|@\^:\ Magic incantation to repeat the leading bit 65 times. Works something like: make all bits zero, index into the array of bits, prepend that result to the bits.
64 2# Reshape the length 128 array into a matrix with two columns.
2/ Convert both columns from base 2.