  The cyclic insertion conjecture of Borwein, Bradley, Broadhurst and
Lison\v{e}k states that inserting all cyclic shifts of some fixed blocks of 2's
into the multiple zeta value {\zeta}(1,3,...,1,3) gives an explicit rational
multiple of a power of {\pi}. In this paper we use motivic multiple zeta values
to establish a non-explicit symmetric insertion result: inserting all possible
permutations of some fixed blocks of 2's into {\zeta}(1,3,...,1,3) gives some
rational multiple of a power of {\pi}.
