Do you get synesthesia from LSD?

It is important to distinguish between true synesthesia and what we call synesthesia-like perception. True synesthesia is a permanent neurological condition in which sensory signals automatically activate other sensory modalities. With LSD use, we are dealing with something more accurately called a synesthesia-like experience—it is temporary and caused by the heightened brain connectivity described by Marcel.
What Marcel mentions regarding cross-activation between cortices is precise. On LSD, we see a great deal of communication between the visual cortex and other areas—not just auditory, but also emotional-sensory. People report hearing colors in music, but also feeling shapes and movements as emotions, smells as colors, etc. This is because LSD increases the network integration of your brain.
An additional important mechanism: psilocybin and LSD also reduce the activity of the default mode network (DMN)—that is, the network that is active when you are not focused on external tasks. This DMN normally helps maintain the separation between different sensory modalities. When this network is less active, that separation blurs, causing synesthesia-like experiences.
Also interesting: not everyone who takes LSD experiences synesthesia. This is related to individual differences in receptor occupancy, genetic variations in neurotransmitter systems, and also simply how sensitive someone is to these types of perceptual shifts. Some people report strong synesthesia-like experiences at low doses, while others do not have this even at high doses.
Another point: time plays a role. Synesthesia-like experiences are usually most intense during the peak of the LSD experience (around 4-6 hours after ingestion) and diminish as the drug wears off.
Finally, one more thing: synesthesia-like experiences can be very instructive because they show how flexible and interpretable your perception actually is. For creative processes (music, art), these types of experiences can be inspiring, but it is also good to keep in mind that these are temporary perceptual shifts, not permanent changes in your brain.
Yes, LSD can cause synesthesia-like experiences, especially at higher doses. This happens because LSD increases communication between different brain regions. However, it is usually temporary and varies greatly from person to person, depending on sensitivity and genetic factors. Not everyone experiences it, even at the same dosage.