Can't access Google Cloud SQL instance from different GCP project, despite setting IAM permissions
Figured it out eventually - perhaps this will be useful to someone else encountering the same problem. 
Problem:

The problem was that the "Cloud SQL Editor" role is not a superset
  of the "Cloud SQL Client", as I had imagined; "Cloud SQL Editor"
  allows administration of the Cloud SQL instance, but doesn't allow
  basic connectivity to the database.

Solution:

Deleting the IAM entry granting Cloud SQL Editor permissions and
  replacing it with one granting Cloud SQL Client permissions fixed the
  issue and allowed the database connection to go through.
I'm attempting to access a Google Cloud SQL instance stored on one Cloud Platform project from an App Engine application on another project, and it's not working.
Connections to the SQL instance fail with this error:
OperationalError: (2013, "Lost connection to MySQL server at 'reading initial communication packet', system error: 38")
I followed the instructions in Google's documentation and added the App Engine service account for the second project to the IAM permissions list for the project housing the Cloud SQL instance (with "Cloud SQL Editor" as the role). The connection details and configuration I'm using in my app are identical to those being used in a perfectly functioning App Engine app housed in the same project as the Cloud SQL instance.
The only thing that seems off about my configuration is that in my second GCP project, while an App Engine service account that looks like the default one ([MY-PROJECT-NAME]@appspot.gserviceaccount.com) appears in the IAM permissions list, this service account is not listed under the Service Accounts tab of IAM & Admin. The only service account listed is the Compute Engine default service account. I haven't deleted any service accounts; there's never been an App Engine default service account listed here, but apart from the MySQL connection the App Engine app runs fine.
Not sure if it's relevant, but I'm running a Python 2.7 app on the App Engine Standard Environment, connecting using MySQLdb.