It works!
julia> s = Symmetric(mat)
6×6 Symmetric{Float64,Array{Float64,2}}:
1.0 0.367879 0.0183156 0.00012341 1.12535e-7 1.38879e-11
0.367879 1.0 0.367879 0.0183156 0.00012341 1.12535e-7
0.0183156 0.367879 1.0 0.367879 0.0183156 0.00012341
0.00012341 0.0183156 0.367879 1.0 0.367879 0.0183156
1.12535e-7 0.00012341 0.0183156 0.367879 1.0 0.367879
1.38879e-11 1.12535e-7 0.00012341 0.0183156 0.367879 1.0
julia> inv(s)
6×6 Symmetric{Float64,Array{Float64,2}}:
1.18147 -0.502645 0.1883 -0.0692716 0.0250252 -0.0079607
-0.502645 1.39526 -0.582587 0.217304 -0.0786496 0.0250252
0.1883 -0.582587 1.42474 -0.59216 0.217304 -0.0692716
-0.0692716 0.217304 -0.59216 1.42474 -0.582587 0.1883
0.0250252 -0.0786496 0.217304 -0.582587 1.39526 -0.502645
-0.0079607 0.0250252 -0.0692716 0.1883 -0.502645 1.18147
julia> issymmetric(inv(s))
true
Hooray for special matrix types!