Are probabilities of mutations symmetric?
The short answer is no, there is heterogeneity in the rate at which different nucleotides mutate into one another. This is generally a property of their differing chemistries (although I'm not an expert on this). Therefore, it doesn't really make sense to talk of alleles being $p$ or  $q$ like we do in most of population genetics, because the actual nucleotide (whether it's A, C, G or T does make a difference).
This is important in fields like phylogenetics, where people construct substitution matrices which describes the rate at which one base in a sequence changes to another nucleotide. Currently, it's possible to estimate the matrix parameters from empirically data relatively easily.
For example, in one of the earliest and most simple models, Kimura (1980) introduced a matrix which had two parameters - one for the mutation rate of transition substitutions (A/C -&gt; G/T, more likely) and one for the rate of transversions (A/G -&gt; T/C, less likely). Later methods got progressively more complex, which account for e.g. the different amino/keto properties of different nucleotides. Felsenstein's model (1981) accounted for the equilibrium frequency of the target nucleotide. These substitution values are also allowed to vary across time as well within a phylogenetic tree (e.g. Yang 1994).

References

Kimura, Motoo. &quot;A simple method for estimating evolutionary rates of base substitutions through comparative studies of nucleotide sequences.&quot; Journal of molecular evolution 16.2 (1980): 111-120.
Felsenstein, Joseph. &quot;Evolutionary trees from DNA sequences: a maximum likelihood approach.&quot; Journal of molecular evolution 17.6 (1981): 368-376.
APA
Yang, Ziheng. &quot;Maximum likelihood phylogenetic estimation from DNA sequences with variable rates over sites: approximate methods.&quot; Journal of Molecular evolution 39.3 (1994): 306-314.