Why is it ok to mix up the order of the time series in a moving block bootstrap?

For a time series consisting of a sequence of observations $x_1, x_2, \dots, x_n$, the moving block bootstrap (See here and wikipedia) is implemented thus (emphasis mine):

1. Pick a block length $k$.
2. Create $n - k + 1$ overlapping blocks of length $k$, so that the first block $x_1^k$ consists of the subsequence $(x_1, \dots, x_k)$, the second $x_2^{k+1}$ is the subsequence $(x_2, \dots, x_{k+1})$, etc.
3. Sample $m = \operatorname{round}(\frac{n}{k})$ blocks with replacement.
4. The bootstrapped observations are then obtained by aligning these blocks back to back, in the order they were picked.

Unlike the regular non-parametric bootstrap, in which the sample order does not make a difference, the moving block bootstrap changes the original chronological ordering of the time series. In the extreme case, where the block length $k = 1$, we can obtain the original sequence in reverse order when $(x_n, \dots, x_1)$ is the bootstrap sample.

This seems counter-intuitive to me. Is there an intuitive explanation on why it is ok to mix up the ordering of the blocks?

It is sensitive to the choice of $k$. You would choose $k$ large enough to capture the important lags (though this isn't the only consideration, there's several tradeoffs come into the choice). So each block provides its own picture of (and so information to estimate) the time-series characteristics like autocorrelation structure.
Certainly it shuffles the order across the blocks, but if $k$ is not too small (and the series is stationary) that effect should be relatively small compared to the information within-blocks. If you had any serial dependence, you would not choose $k=1$.