The only things working are posts and pages, when translated, which means that the language switcher knows the correct path to translated posts or pages.
Looking to the request/response, the correct language is requested (en_EN) but the response has the default language (pt_PT) as shown below (setting the cookie):
Requests on HTTP, to /en
Remote Address:10.10.10.10:80
Request URL:https://MYSITE.COM/en/
Request Method:GET
Status Code:200 OK
Request Headersview parsed
GET /en/ HTTP/1.1
Host: MYSITE.COM
Connection: keep-alive
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,image/webp,*/*;q=0.8
User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_10_1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/38.0.2125.122 Safari/537.36
DNT: 1
Referer: https://MYSITE.COM/
Accept-Encoding: gzip,deflate,sdch
Accept-Language: en-US,en;q=0.8
Cookie: PHPSESSID=3l3d4sqferc20torja9ed6os54; cookie_notice_accepted=true; pll_language=pt; wordpress_test_cookie=WP+Cookie+check; w3tc_logged_out=
Response Headersview source
Cache-Control:no-store, no-cache, must-revalidate, post-check=0, pre-check=0
Connection:keep-alive
Content-Encoding:gzip
Content-Type:text/html; charset=UTF-8
Date:Wed, 19 Nov 2014 16:19:45 GMT
Expires:Thu, 19 Nov 1981 08:52:00 GMT
Link:<https://MYSITE.COM/wp-json>; rel="https://github.com/WP-API/WP-API"
Pragma:no-cache
Server:nginx/1.0.15
Set-Cookie:pll_language=en; expires=Thu, 19-Nov-2015 16:19:43 GMT; Max-Age=31536000; path=/; domain=.MYSITE.COM
Transfer-Encoding:chunked
Vary:Accept-Encoding
X-Pingback:https://MYSITE.COM/xmlrpc.php
——————————————————————
Requests on HTTPS, to /en
Remote Address:10.10.10.10:443
Request URL:https://MYSITE.COM/en/
Request Method:GET
Status Code:200 OK
Request Headersview parsed
GET /en/ HTTP/1.1
Host: MYSITE.COM
Connection: keep-alive
Accept: text/html,application/xhtml+xml,application/xml;q=0.9,image/webp,*/*;q=0.8
User-Agent: Mozilla/5.0 (Macintosh; Intel Mac OS X 10_10_1) AppleWebKit/537.36 (KHTML, like Gecko) Chrome/38.0.2125.122 Safari/537.36
DNT: 1
Referer: https://MYSITE.COM/
Accept-Encoding: gzip,deflate,sdch
Accept-Language: en-US,en;q=0.8
Cookie: PHPSESSID=3l3d4sqferc20torja9ed6os54; cookie_notice_accepted=true; wp-settings-14=hidetb%3D1%26editor%3Dhtml%26posts_list_mode%3Dlist%26libraryContent%3Dbrowse; wp-settings-time-14=1416395461; wordpress_test_cookie=WP+Cookie+check; w3tc_logged_out=; pll_language=pt
Response Headersview parsed
HTTP/1.1 200 OK
Server: nginx/1.0.15
Date: Wed, 19 Nov 2014 16:21:22 GMT
Content-Type: text/html; charset=UTF-8
Transfer-Encoding: chunked
Connection: keep-alive
Vary: Accept-Encoding
Expires: Thu, 19 Nov 1981 08:52:00 GMT
Cache-Control: no-store, no-cache, must-revalidate, post-check=0, pre-check=0
Pragma: no-cache
X-Pingback: https://MYSITE.COM/xmlrpc.php
Link: <https://MYSITE.COM/wp-json>; rel="https://github.com/WP-API/WP-API"
Content-Encoding: gzip