Let's have a listen to the first movement. Here is an excellent recording by Gilels, Kogan and Rostropovich made in 1959. The first movement goes to the 7:38 mark:
Here is a site where you can download a pdf of the score. The first movement begins with a fugal texture, but with the cello stating the theme in extremely high harmonics:
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Then the violin comes in with the theme as the cello continues:
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When the piano enters with the theme it is in the low bass in octaves:
As you can see, this is not dissonant, far less dissonant than late 19th century music! The cello statement of the theme is on A or rather the Aeolian mode. When the violin enters, though following the same basic intervals, it is in G major. Then the entry of the piano is in Phrygian mode on B. Though the texture is that of a fugue with the three instruments each stating the theme, harmonically this is very unlike a traditional fugue in which the voices would typically enter on the tonic, dominant and then tonic again. Here they enter on A, G and B with a modal structure.
The fugal opening serves as an introduction to a movement in sonata form, moderato, that begins just after the 3 minute mark in the clip above. This begins in E minor with the piano stating another version of that opening fugue theme. At the 5 minute mark a new section begins, marked Poco più mosso with a new theme in G major:
After the stating of these two themes, the exposition of a sonata form, there is a development section:
And recapitulation of both themes. Here is the beginning of the recapitulation of the opening E minor theme:
As an example of how he might alter a cadence, here is how the first movement ends:
There are some elements of a typical cadence: the D sharp leading tone in the bass which together with the B and F sharp in the upper voices gives a full dominant triad. But no seventh and traditionally, an authentic cadence requires the dominant to be in root position, not first inversion. But then the D sharp is contradicted by a D natural in the next measure and the tonic chord is rendered rather doleful by the reiterated F natural decoration for four measures! The E minor cadence is essentially combined with a Phrygian cadence.
Let's listen to this first movement again in a different performance. Here is the trio of Piano: Martha Argerich, Violin: Gidon Kremer, Violoncello: Mischa Maisky: