Following up on Anthony Quas's post, let $g(x) = 4x(1-x)$ and $h(x) = \sin(\pi x)$.  The equations $g^{(n)}(x) = 1/2$ and $h^{(n)}(x) = 1/2$ each have $2^n$ roots in $[0,1]$. If we sort those roots in order as $y_1 < y_2 < \cdots < y_{2^n}$ and $z_1 < z_2 < \cdots < z_{2^n}$, we should have $f(y_i) = z_i$. 

Here is a plot, and list of values, for $n=6$:

[![enter image description here][1]][1]

    0.0002     0.0005
    0.0014     0.0033
    0.0038     0.0079
    0.0074     0.0132
    0.0121     0.0211
    0.0181     0.0282
    0.0252     0.0369
    0.0335     0.0465
    0.0429     0.0604
    0.0534     0.0716
    0.0650     0.0832
    0.0776     0.0942
    0.0912     0.1092
    0.1058     0.1220
    0.1214     0.1375
    0.1379     0.1546
    0.1552     0.1790
    0.1734     0.1974
    0.1924     0.2153
    0.2121     0.2310
    0.2325     0.2505
    0.2536     0.2659
    0.2752     0.2833
    0.2974     0.3012
    0.3201     0.3262
    0.3432     0.3454
    0.3666     0.3651
    0.3904     0.3835
    0.4145     0.4082
    0.4388     0.4292
    0.4632     0.4543
    0.4877     0.4813
    0.5123     0.5187
    0.5368     0.5457
    0.5612     0.5708
    0.5855     0.5918
    0.6096     0.6165
    0.6334     0.6349
    0.6568     0.6546
    0.6799     0.6738
    0.7026     0.6988
    0.7248     0.7167
    0.7464     0.7341
    0.7675     0.7495
    0.7879     0.769
    0.8076     0.7847
    0.8266     0.8026
    0.8448     0.821
    0.8621     0.8454
    0.8786     0.8625
    0.8942     0.878
    0.9088     0.8908
    0.9224     0.9058
    0.9350     0.9168
    0.9466     0.9284
    0.9571     0.9396
    0.9665     0.9535
    0.9748     0.9631
    0.9819     0.9718
    0.9879     0.9789
    0.9926     0.9868
    0.9962     0.9921
    0.9986     0.9967
    0.9998     0.9995

  [1]: https://i.sstatic.net/EqHm0.png