WCF service returns 404 over https but not http

I'm migrating an existing service from HTTP (Dev/UAT) to HTTPS (Production), and I'm having trouble with the configuration. Here is the system.serviceModel section of my web.config:

<system.serviceModel>
    <behaviors>
      <serviceBehaviors>
        <behavior name="">
          <serviceMetadata httpGetEnabled="true" />
          <serviceDebug includeExceptionDetailInFaults="true" />
        </behavior>
      </serviceBehaviors>
    </behaviors>
    <serviceHostingEnvironment multipleSiteBindingsEnabled="false" />
    <services>
      <service name="MyService">
        <endpoint name="MyEndpoint" address="" binding="wsHttpBinding"
            bindingConfiguration="secureBinding" contract="IMyService" />
      </service>
    </services>
    <bindings>
      <wsHttpBinding>
        <binding name="secureBinding">
          <security mode="Transport"></security>
        </binding>
      </wsHttpBinding>
    </bindings>
</system.serviceModel>

I've tried this using both basicHttpBinding and wsHttpBinding, with the same results:

• I can call the service from my SOAP client using http://server.domain.com/MyService.svc
• I can hit the service from a browser using https://server.domain.com/MyService.svc
• I can't call the service from my SOAP client using https://server.domain.com/MyService.svc - the call always errors with 404: not found.

My https site is certified using a certificate that was issued by a CA on the corporate domain, and I've verified that I have that CA's certificate installed in Trusted Root Certification Authorities on the system from which I'm making the calls.

The relevant client code:

Service service = new Service();
s